Plate tectonics: Evidence of plate movement

WEBVTT
Kind: captions
Language: en

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Let's think a little bit
about the different clues that

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have led us to conclude that we
have these lithospheric plates

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moving relative to each other.

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Now, the first clue,
and this is something

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that I think many students
even in elementary school

00:00:15.430 --> 00:00:18.310
first experience when they
first learn about geography,

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is that it looks like the
continents could kind of fit

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into each other.

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And the most
obvious one of these

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is when you look at this kind
of little pointy part of South

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America, and if you
have a more detailed map

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it really is
amazing, how well it

00:00:30.500 --> 00:00:35.430
seems to fit into the Nigerian
Basin right here in Africa.

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It looks like at one time
this little pointy part

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was nudged into
this part of Africa,

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that they were
actually connected.

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And if you're a little
bit more creative,

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there are other
parts of the world

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that you can kind of start to
see how they might have fit

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in with each other in the past.

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And that by itself, that's
just a very small clue,

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but it kind of hints at well,
maybe if they at one time

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were fitting next to each other,
if this was kind of connected,

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then they've had to
moved apart at some time.

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Although it doesn't tell
us that it's still moving

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or what might have
caused the movement.

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And it definitely
doesn't definitively

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tell us that they even moved.

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Maybe this is just a
coincidence that this coast

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of South America looks
very similar to this coast

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right here of Africa.

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Now, the next clues
truly came over

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I would say about the
last 60 or 70 years.

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The first clue is
that OK, if you

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go to the mid-Atlantic
ridge right here--

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So if you look at
the Atlantic Ocean.

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Let me look at this
photograph right over here.

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You don't normally see the
oceans highlighted like this.

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So let me make it
very clear to you.

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This right here
is South America.

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This right here is Africa.

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This right here
is North America.

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And so if you actually
look at the elevations

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in the middle of the
ocean people noticed

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in the middle of the
20th century that gee,

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there's a ridge in the
middle of the Atlantic Ocean.

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There's kind of a
mountain range that

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goes straight up the middle
of the Atlantic Ocean.

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So that by itself
doesn't tell you

00:02:02.660 --> 00:02:06.030
that you have these plates
that are moving apart,

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but it is kind of a
curious thing to look at.

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And not only is
there this ridge.

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There's lot of underwater
volcanic activity.

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You have magma flowing
out and lava flowing out

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into the water, and it's kind of
forming this ridge that really

00:02:18.550 --> 00:02:20.472
goes across the
whole Atlantic Ocean.

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There are other
ridges in the world

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like that, underwater ridges.

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You have one over here
in the Pacific Ocean.

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You have these here
in the Indian Ocean.

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That's just a little
clue, but that by itself

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doesn't tell you that
these plates are actually

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moving apart at the ridge.

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The more conclusive--
this is just

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the beginning of
the clue-- but what

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made this conclusive is
one, the separate discovery.

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And this is what's
interesting is

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that you have these
separate discoveries

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in different domains
that eventually

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let you come to a
pretty neat conclusion.

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So you've had a separate
discovery that if you

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look at different
eras of magnetic rock,

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or maybe I should say magnetic
rock from different periods

00:03:04.140 --> 00:03:05.441
in geologic time.

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And you can tell where
they are in geologic time

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by how they're layered.

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So this would be newer rock.

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And then this would
be a little bit older.

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And then this would
be even, even older.

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Geologists noticed
something interesting.

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If I were to take magnetic
rock, and if it was molten lava,

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and if it were to harden,
remember it's magnetic rock

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so it would want to align
with the poles the same way

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a compass would.

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So if I had a bunch
of magnetic-- so

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let's say this is
some lava right here.

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And so the molecules
can align themselves.

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Since it's a liquid and
they can align themselves,

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they are going to naturally
want to align with the poles.

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So they'll naturally all want to
align in one direction because

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of Earth's magnetic field.

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And so when that lava
hardens into actual rock

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that alignment will
kind of be frozen.

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Now, if Earth's magnetic
field was constant over time,

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then when you look at magnetic
rocks from any period,

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you would expect them all to be
aligned in the same direction.

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So since we're taking a cross
section of rock here let's

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say an alignment towards the
North pole looks like this.

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And I draw it like that.

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That's kind of an arrow
pointing into our screen.

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And let's say an alignment
pointing to the South pole

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would look like this.

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This would be an arrow
pointing out of our screen.

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So what you would
expect is the newer

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rock that kind of the alignment,
the field, the alignment

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of the rock, would go into
the screen, and then the older

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rock, it would still go
older into the screen.

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So if I were to
draw a top view--

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Let me draw it
like this just so I

00:04:51.930 --> 00:04:53.763
make sure that everyone
is on the same page.

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So let me just draw a
cross section like this

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so that we know what
we're talking about.

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And so this is the
surface up here.

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This up here is the surface.

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When I talk about
going into the page

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that means that
the magnetic rock

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would be aligned
in that direction.

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And when I talk about
going out of the page

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it means, so if
I were to draw it

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like that, that means that
the magnetic rock would

00:05:21.810 --> 00:05:23.030
be aligned in that direction.

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Now, like I said, if the
magnetic field of Earth

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never changed, then lava
that essentially cools down

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into non-lava rock, or you
can say freezes into rock,

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it would all point
into the same direction

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regardless of when it hardened.

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This would be the situation
in a constant magnetic field.

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But what we've seen is
that that's not the case.

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When you look at
older magnetic rock,

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and depending on
how old you go, you

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have the newer
rock that's aligned

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with our current magnetic field.

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You go a little bit
older, and right now we

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think it's about 780,000
years ago roughly.

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You have to find rock of
that age, magnetic rock that

00:06:03.420 --> 00:06:04.800
hardened at that time.

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It's actually in the
opposite direction.

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So actually, the magnetic
rock has hardened in a way

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so it's as if the North Pole
was at the South Pole now,

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the magnetic North pole.

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So it's aligned in the
opposite direction.

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So it's kind of pointing
out of the page here.

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And then if you get
even older rock,

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it's more aligned with
our traditional direction.

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So it's more aligned than that.

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And so the only
reasonable conclusion

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that we can draw from this is
that Earth's magnetic field

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has actually
fluctuated over time.

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Now, you're probably
thinking, Sal,

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how is this relevant
to plate tectonics?

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Well, once you accept that
magnetic fields fluctuate

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over the history of
the Earth, there's

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another interesting
observation you

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can make about the
rock that's kind

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of at the basin of
the ocean floor.

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So not only do you have
this mid-Atlantic ridge,

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you have these volcanoes
spewing kind of new rock

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into the ocean, creating this
kind of underwater mountain

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ridge, but it also turns
out that the rock that

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forms the sea
floor also contains

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a lot of magnetite,
which is magnetic.

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And what's really interesting
about that-- so let me draw.

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So we're going to
have a top view

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just like we have over here.

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So let's say this is the
mid-Atlantic ridge right here.

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Now, this is really cool.

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So when they look
at rocks that are

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very close to the mid-Atlantic
ridge, they're aligned--

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and once again, we're
looking at rocks

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at the floor of the ocean--
they're aligned in a way

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that you would expect with
the current magnetic field.

00:08:03.930 --> 00:08:06.675
They are aligned just
like that, the way

00:08:06.675 --> 00:08:08.050
that you would
expect when you're

00:08:08.050 --> 00:08:10.410
looking at the magnetic rock
that's close to the ridge.

00:08:13.452 --> 00:08:14.910
But if you go a
little bit further,

00:08:14.910 --> 00:08:17.201
and when I say a little bit
I'm talking about thousands

00:08:17.201 --> 00:08:19.240
of miles, but when you
go further out from that

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you have stripes of
other magnetic rock

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that is going in the
opposite direction.

00:08:27.530 --> 00:08:28.565
It's going like this.

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And what's even
cooler than the idea

00:08:35.090 --> 00:08:37.760
that it's switched directions
depending on how far you've

00:08:37.760 --> 00:08:39.360
gone from the rift
is that there's

00:08:39.360 --> 00:08:44.100
a symmetric stripe of
magnetic rock on exactly

00:08:44.100 --> 00:08:46.780
the same distance, or roughly
the same distance away

00:08:46.780 --> 00:08:48.627
from the rift
that's also pointing

00:08:48.627 --> 00:08:49.585
in that same direction.

00:08:54.522 --> 00:08:55.980
And you go a little
bit further out

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and you'll find some
rock that's pointing

00:08:57.688 --> 00:08:58.860
in the original direction.

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And even better, you go on
the symmetric other side

00:09:04.270 --> 00:09:06.860
of the actual ridge and
you find another set

00:09:06.860 --> 00:09:12.220
of rocks that's doing
the exact same thing.

00:09:12.220 --> 00:09:16.170
So if you accept that Earth's
magnetic field has kind of been

00:09:16.170 --> 00:09:20.100
flip flopping over time, the
only reasonable conclusion,

00:09:20.100 --> 00:09:23.420
at least that I could think of,
or the geologists can think of,

00:09:23.420 --> 00:09:28.630
is that sure, all
of this was formed

00:09:28.630 --> 00:09:30.390
at a similar period in time.

00:09:30.390 --> 00:09:32.760
This came out as
lava, magnetic lava,

00:09:32.760 --> 00:09:35.570
and then it all aligned
with Earth's magnetic field.

00:09:35.570 --> 00:09:37.320
And that's why it looks similar.

00:09:37.320 --> 00:09:41.497
You fast forward in time
some and the only way--

00:09:41.497 --> 00:09:43.330
Or actually, let's not
fast forward in time.

00:09:43.330 --> 00:09:45.660
The only way that these
could have formed,

00:09:45.660 --> 00:09:47.210
and they could have
been so similar.

00:09:47.210 --> 00:09:49.310
So if we rewind in
time, the only way

00:09:49.310 --> 00:09:52.930
that these purple magnetic rocks
could have aligned this way

00:09:52.930 --> 00:09:55.430
in exactly the same way and
exactly the same distance

00:09:55.430 --> 00:09:58.000
is if at some point they were
much closer to each other,

00:09:58.000 --> 00:09:59.333
if they were actually connected.

00:09:59.333 --> 00:10:04.240
So if we rewind in time maybe
at the mid-Atlantic rift

00:10:04.240 --> 00:10:09.070
you had all of the
purple rock coming out

00:10:09.070 --> 00:10:14.750
from those underwater
volcanoes, and at that time,

00:10:14.750 --> 00:10:18.470
Earth's magnetic field was the
opposite as it is right now.

00:10:18.470 --> 00:10:22.420
And then of course,
you had this blue rock

00:10:22.420 --> 00:10:25.660
that is looking like that.

00:10:25.660 --> 00:10:27.680
And so this seems like a
reasonable explanation.

00:10:27.680 --> 00:10:30.260
This rock and this rock
were at some point touching.

00:10:30.260 --> 00:10:32.670
They were actually
formed at the exact place

00:10:32.670 --> 00:10:35.350
and at the same time.

00:10:35.350 --> 00:10:38.080
And so if this is the
case, if at one point

00:10:38.080 --> 00:10:39.790
this purple rock
was all together,

00:10:39.790 --> 00:10:42.820
and they formed at the same
time at the mid-Atlantic rift,

00:10:42.820 --> 00:10:44.530
we're assuming all
the rock was-- well,

00:10:44.530 --> 00:10:45.550
we don't have to make
that assumption--

00:10:45.550 --> 00:10:48.110
but if you assume that they've
formed at the same time,

00:10:48.110 --> 00:10:50.920
and that based on the pattern it
really does look like they do,

00:10:50.920 --> 00:10:54.347
and it's a symmetric
distance away from that rift,

00:10:54.347 --> 00:10:56.430
then the only reasonable
conclusion I can think of

00:10:56.430 --> 00:10:58.910
is that the rift has
had to move apart.

00:10:58.910 --> 00:11:01.410
The rift has had to move
apart from this period

00:11:01.410 --> 00:11:02.610
to that period.

00:11:02.610 --> 00:11:05.540
And there was a time when all
this blue rock was together.

00:11:05.540 --> 00:11:08.940
So that by itself, that frankly
is the most definitive evidence

00:11:08.940 --> 00:11:11.730
in the 1960s where it kind of
became conclusive that you did

00:11:11.730 --> 00:11:15.440
have these plates that were
moving away from each other.

00:11:18.742 --> 00:11:21.200
And obviously if the plates
are moving away from each other

00:11:21.200 --> 00:11:24.482
at some point, and that means
that just based on the way

00:11:24.482 --> 00:11:26.190
the map looks, at some
point they're also

00:11:26.190 --> 00:11:27.700
going to be moving
into each other.

00:11:27.700 --> 00:11:30.020
We could talk more about
that in future videos.

00:11:30.020 --> 00:11:33.070
But, you know, at
certain points one plate

00:11:33.070 --> 00:11:34.590
is moving under another.

00:11:34.590 --> 00:11:37.007
And we'll talk about how
that might partially explain,

00:11:37.007 --> 00:11:38.840
and we'll talk about
all of the explanations

00:11:38.840 --> 00:11:41.422
for why we think the plates
might actually be moving.

00:11:41.422 --> 00:11:43.630
But now, if we fast forward
to more present times now

00:11:43.630 --> 00:11:46.140
that we have GPS satellites
and all the rest,

00:11:46.140 --> 00:11:49.060
we can actually measure
the movement of the plates.

00:11:49.060 --> 00:11:52.500
This is actually
an image from NASA

00:11:52.500 --> 00:11:54.620
showing the vector
of the movements

00:11:54.620 --> 00:11:57.500
at different points on
the surface of the planet.

00:11:57.500 --> 00:11:59.750
And you can see we've
gotten a lot of vectors

00:11:59.750 --> 00:12:02.230
from the United States, so
it's almost hard to read

00:12:02.230 --> 00:12:04.385
since it's so chalk
full of vectors.

00:12:04.385 --> 00:12:06.620
But you can see right
over here in Hawaii.

00:12:06.620 --> 00:12:09.830
The Pacific Plate at that point
is moving in this Northwest

00:12:09.830 --> 00:12:12.800
direction as measured
by GPS satellites.

00:12:12.800 --> 00:12:16.050
And I want to make clear, this
movement is relatively slow.

00:12:16.050 --> 00:12:18.624
It's roughly the speed at
which fingernails grow,

00:12:18.624 --> 00:12:20.290
but if you do it over
millions of years,

00:12:20.290 --> 00:12:23.470
that actually amounts
to thousands of miles.

00:12:23.470 --> 00:12:26.422
So we're talking on the order
of about a centimeter a year

00:12:26.422 --> 00:12:27.380
for most of the plates.

00:12:27.380 --> 00:12:29.671
Some of the plates might be
moving a little bit faster,

00:12:29.671 --> 00:12:31.552
maybe close to 10
or 15 centimeters,

00:12:31.552 --> 00:12:33.135
but most are moving
about a centimeter

00:12:33.135 --> 00:12:35.332
a year, at the same rate
your fingernails are going.

00:12:35.332 --> 00:12:37.790
But this is fascinating because
we can actually measure it,

00:12:37.790 --> 00:12:39.200
because GPS is so accurate.

00:12:39.200 --> 00:12:41.200
Over here it looks like
the North American plate

00:12:41.200 --> 00:12:45.580
is kind of rotating
generally in that direction.

00:12:45.580 --> 00:12:50.320
The Nazca Plate right
here is moving roughly

00:12:50.320 --> 00:12:54.129
in that direction, moving
into the South American Plate.

00:12:54.129 --> 00:12:55.420
I'll leave you there right now.

00:12:55.420 --> 00:12:57.931
And actually before
I leave you there,

00:12:57.931 --> 00:12:59.430
this is another
thing that I got off

00:12:59.430 --> 00:13:01.638
the Wikipedia that shows
that same magnetic striping.

00:13:01.638 --> 00:13:03.670
It's maybe a slightly
neater drawing.

00:13:03.670 --> 00:13:06.566
I don't know which one might
be more helpful for you.

00:13:06.566 --> 00:13:08.190
But I'll leave you
there in this video.

00:13:08.190 --> 00:13:10.606
In the next video, we'll think
about some of the theories.

00:13:10.606 --> 00:13:12.317
We know now that the
plates are moving.

00:13:12.317 --> 00:13:13.900
Let's think about
some of the theories

00:13:13.900 --> 00:13:17.480
as to why they might
actually be moving.


Salman Khan on KQED MindShift

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:01.530
My name's Sal Khan.

00:00:01.530 --> 00:00:04.550
I'm the founder and faculty
of the Khan Academy,

00:00:04.550 --> 00:00:09.276
and it's most known for a
library of 2,000 plus videos

00:00:09.276 --> 00:00:10.650
on everything from
basic addition

00:00:10.650 --> 00:00:13.404
all the way up to calculus
and biology and chemistry.

00:00:13.404 --> 00:00:14.820
But what we're
really trying to do

00:00:14.820 --> 00:00:18.300
is build a free, not for
profit virtual school

00:00:18.300 --> 00:00:19.650
that anybody can go.

00:00:19.650 --> 00:00:21.650
There are the videos, but
there's also software.

00:00:21.650 --> 00:00:23.460
They can start at
basic concepts and then

00:00:23.460 --> 00:00:26.630
just keep moving up until
they know calculus and physics

00:00:26.630 --> 00:00:28.610
and whatever else.

00:00:28.610 --> 00:00:30.685
The details have yet
to be worked out.

00:00:30.685 --> 00:00:32.060
But what we're
building-- we talk

00:00:32.060 --> 00:00:33.240
that it's a virtual
school, but we actually

00:00:33.240 --> 00:00:34.656
do see it as kind
of the operating

00:00:34.656 --> 00:00:36.444
system for the
school of the future,

00:00:36.444 --> 00:00:37.860
and we're starting
trials already.

00:00:37.860 --> 00:00:39.526
And the model we're
experimenting with--

00:00:39.526 --> 00:00:41.310
and it seems to be
working-- is you

00:00:41.310 --> 00:00:43.410
use the self-paced software.

00:00:43.410 --> 00:00:44.640
You use the video.

00:00:44.640 --> 00:00:48.260
You use the technology
for every student

00:00:48.260 --> 00:00:49.640
to work at their own pace.

00:00:49.640 --> 00:00:52.344
So literally you have a class
of-- you could have 30 students

00:00:52.344 --> 00:00:54.010
or 50 students or
however many students,

00:00:54.010 --> 00:00:55.790
and they actually don't all
have to be the same age.

00:00:55.790 --> 00:00:57.498
Preferably they're
all of different ages,

00:00:57.498 --> 00:00:59.156
all of different skill levels.

00:00:59.156 --> 00:01:01.030
And they use something
like the Khan Academy.

00:01:01.030 --> 00:01:02.680
They all work at their own pace.

00:01:02.680 --> 00:01:04.769
And the teacher now doesn't
have to give lectures,

00:01:04.769 --> 00:01:05.730
doesn't have to grade
homework, doesn't

00:01:05.730 --> 00:01:08.521
have to do anything other
than-- they have a real time

00:01:08.521 --> 00:01:09.020
dashboard.

00:01:09.020 --> 00:01:11.353
We actually have this already,
but they have a real time

00:01:11.353 --> 00:01:13.600
dashboard where they see
where every student is,

00:01:13.600 --> 00:01:14.830
working at their own pace.

00:01:14.830 --> 00:01:16.550
And it just highlights
the kids that are stuck.

00:01:16.550 --> 00:01:18.115
Not the kids that are behind or
the kids that are ahead, just

00:01:18.115 --> 00:01:19.114
the kids that are stuck.

00:01:19.114 --> 00:01:21.870
They've already done 100
computer-generated exercises.

00:01:21.870 --> 00:01:24.300
Most students who've done
100 exercises have moved on.

00:01:24.300 --> 00:01:25.490
This student hasn't.

00:01:25.490 --> 00:01:28.210
And the model is-- as a
first point of intervention,

00:01:28.210 --> 00:01:29.960
actually, the teacher
or even the software

00:01:29.960 --> 00:01:32.168
can say, why don't you pair
up those two students who

00:01:32.168 --> 00:01:34.120
are stuck on negative
numbers, why don't you

00:01:34.120 --> 00:01:37.830
pair them up with Susie who's
already gotten through that

00:01:37.830 --> 00:01:39.110
and have her teach them.

00:01:39.110 --> 00:01:40.860
And then you have this
whole peer to peer,

00:01:40.860 --> 00:01:43.068
when you actually teach
something you learn it better

00:01:43.068 --> 00:01:45.730
and scales up the whole-- it
makes it a social experience.

00:01:45.730 --> 00:01:48.074
But if that doesn't work or
if Susie's not available,

00:01:48.074 --> 00:01:49.240
then the teacher goes there.

00:01:49.240 --> 00:01:51.887
And the teacher does a very
focused, one on one or one

00:01:51.887 --> 00:01:52.720
on two intervention.

00:01:52.720 --> 00:01:53.880
So it changes the model.

00:01:53.880 --> 00:01:57.960
So every minute of a teacher's
time is actually very focused.

00:01:57.960 --> 00:01:59.940
But because it's
very-- you don't

00:01:59.940 --> 00:02:03.650
want kids spending the whole
day on a computer just clicking

00:02:03.650 --> 00:02:05.300
on problems.

00:02:05.300 --> 00:02:08.289
So I'd see you spend
maybe two hours a day

00:02:08.289 --> 00:02:12.040
doing that, to get your
core kind of knowledge

00:02:12.040 --> 00:02:12.740
out of the way.

00:02:12.740 --> 00:02:14.370
You do that, you're going
to do well on the SAT.

00:02:14.370 --> 00:02:15.744
You're going to
pass the AP test.

00:02:15.744 --> 00:02:19.556
You're going to know how
to grammatically write

00:02:19.556 --> 00:02:20.990
a sentence, all of the rest.

00:02:20.990 --> 00:02:22.830
But what it does is it frees
up the rest of the day so you

00:02:22.830 --> 00:02:24.720
can actually do actual
creative projects.

00:02:24.720 --> 00:02:26.420
I mean, there's a lot of talk
about project-based learning.

00:02:26.420 --> 00:02:28.430
But very little of
it is actual, real,

00:02:28.430 --> 00:02:31.010
kind of liberated
creative stuff.

00:02:31.010 --> 00:02:31.510
Yeah.

00:02:31.510 --> 00:02:33.360
I see the school of
the future, big school

00:02:33.360 --> 00:02:35.881
room, every kid working at
their own pace, a lot of time

00:02:35.881 --> 00:02:36.380
freed up.

00:02:36.380 --> 00:02:38.380
And the teacher's now a
coach, and a lot of time

00:02:38.380 --> 00:02:41.570
freed up for kids to
really be creative.


Exponent properties 3

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.610
Simplify x to the
third, and then that

00:00:03.610 --> 00:00:06.410
raised to the fourth
power times x squared,

00:00:06.410 --> 00:00:09.160
and then that raised
to the fifth power.

00:00:09.160 --> 00:00:12.640
Now, here we're going to use
the power property of exponents,

00:00:12.640 --> 00:00:14.090
sometimes called
the "power rule."

00:00:14.090 --> 00:00:17.305
And that just tells us
if I have x to the a,

00:00:17.305 --> 00:00:20.610
and then I raise
that to b-th power,

00:00:20.610 --> 00:00:25.220
this is the same thing as
x to the a times b power.

00:00:25.220 --> 00:00:27.701
And to see why that works,
let's try that with a-- well,

00:00:27.701 --> 00:00:29.200
I won't do it with
these right here.

00:00:29.200 --> 00:00:30.616
I'll do it with a
simpler example.

00:00:30.616 --> 00:00:33.860
Let's say we are
taking x squared

00:00:33.860 --> 00:00:36.515
and then raising that
to the third power.

00:00:36.515 --> 00:00:38.140
Well, in that situation,
that literally

00:00:38.140 --> 00:00:42.080
means multiplying x squared
by itself three times.

00:00:42.080 --> 00:00:44.610
So that literally
means, x squared

00:00:44.610 --> 00:00:48.360
times x squared times x squared.

00:00:48.360 --> 00:00:52.140
I'm taking x squared and I'm
raising it to the third power.

00:00:52.140 --> 00:00:53.910
Now, what does this
mean over here?

00:00:53.910 --> 00:00:55.750
Well, there's a couple
of ways to do it.

00:00:55.750 --> 00:00:57.890
You could just say, look,
I have the same base,

00:00:57.890 --> 00:01:00.780
I'm taking the product, I
can just add the exponents.

00:01:00.780 --> 00:01:03.850
So this is going to be
equal to-- let me do it

00:01:03.850 --> 00:01:07.650
in that same magenta--
this is equal to x to the 2

00:01:07.650 --> 00:01:12.500
plus 2 plus 2 power, or
essentially x to the 3 times 2.

00:01:12.500 --> 00:01:15.810
This right here
is just 3 times 2.

00:01:15.810 --> 00:01:18.692
So we get x to the sixth power.

00:01:18.692 --> 00:01:20.150
And if you want,
you say, hey, Sal,

00:01:20.150 --> 00:01:22.170
I don't understand why you
can add those exponents.

00:01:22.170 --> 00:01:24.836
You just have to remember that x
squared-- this thing right over

00:01:24.836 --> 00:01:27.900
here, we could
rewrite as x times x

00:01:27.900 --> 00:01:30.850
times x times x--
in parentheses,

00:01:30.850 --> 00:01:36.190
I'm putting each of these x
squareds-- times x times x.

00:01:36.190 --> 00:01:38.950
And this is just x
times itself 6 times

00:01:38.950 --> 00:01:41.780
x times x times x times
x times x times x.

00:01:41.780 --> 00:01:44.160
This is just x to
the sixth power.

00:01:44.160 --> 00:01:46.940
So that's why we can add
the exponents like that.

00:01:46.940 --> 00:01:49.210
So let's just use
that power property

00:01:49.210 --> 00:01:51.130
on this expression
right over here.

00:01:51.130 --> 00:01:54.870
We start off with we have
x to the third raised

00:01:54.870 --> 00:01:57.120
to the fourth power.

00:01:57.120 --> 00:02:02.140
So that's just going to be x
to the 3 times 4 power, or x

00:02:02.140 --> 00:02:03.840
to the 12th power.

00:02:03.840 --> 00:02:06.900
Then, we're
multiplying that by x

00:02:06.900 --> 00:02:09.330
squared raised to
the fifth power.

00:02:09.330 --> 00:02:14.350
Well, that's just going to be
x to the 2 times 5 power, or x

00:02:14.350 --> 00:02:16.880
to the 10th power.

00:02:16.880 --> 00:02:19.940
And now we have the same base,
and we're taking the product,

00:02:19.940 --> 00:02:21.530
we can just add the exponents.

00:02:21.530 --> 00:02:24.490
This is going to be equal
to-- this whole expression

00:02:24.490 --> 00:02:27.100
is going to be
equal to x to the 12

00:02:27.100 --> 00:02:32.120
plus 10th power, or
x to the 22nd power.

00:02:32.120 --> 00:02:34.004
And we are done.


Exponent properties 4

WEBVTT
Kind: captions
Language: en

00:00:00.825 --> 00:00:01.975
Simplify 25 a to the third

00:00:01.990 --> 00:00:05.933
and a to the third is being raised to the third power, times b squared

00:00:05.933 --> 00:00:09.723
and all of that over 5 a squared, b times b squared

00:00:09.723 --> 00:00:12.698
So we can do this in multiple ways, simplify different parts.

00:00:12.698 --> 00:00:16.262
What I want to do is simplify this part right over here.

00:00:16.262 --> 00:00:18.990
a to the third power, and we're raising that to the third power.

00:00:18.990 --> 00:00:21.871
So this is going to be from the power property of exponents, or the power rule

00:00:21.871 --> 00:00:27.489
this is going to be the same thing as a to the 3 times 3 power

00:00:27.489 --> 00:00:30.082
So this over here (let me scroll up a little bit)

00:00:30.190 --> 00:00:36.456
is going to be equal to a the 3 times 3 power, or a to the ninth power.

00:00:36.456 --> 00:00:41.329
We could also simplify this b times b squared over here.

00:00:41.329 --> 00:00:45.415
This b times b squared, that is the same thing as b to the first power

00:00:45.415 --> 00:00:49.271
remember, b is just b to the first power. So it's b to the first power times

00:00:49.271 --> 00:00:53.606
b to the second power. So b to the first times b to the second power

00:00:53.606 --> 00:00:59.375
is just equal to b to the one plus two power, which is equal to b to the third power

00:00:59.375 --> 00:01:04.102
and then last thing we could simplify, just right off the bat just looking at this:

00:01:04.102 --> 00:01:08.195
we have a 25 divided by 5. Well that's just going to give us 5.

00:01:08.195 --> 00:01:10.790
or we could say it's going to give us 5 over 1

00:01:10.790 --> 00:01:13.918
if you view it as dividing the numerator and the denominator both by 5.

00:01:13.918 --> 00:01:17.082
So what does our expression simiply to?

00:01:17.082 --> 00:01:25.667
We have 5a to the ninth, and then we still have this b squared here, b squared.

00:01:25.667 --> 00:01:35.200
All of that over a squared times b to the third power... times b to the third power.

00:01:35.200 --> 00:01:36.887
Now, we can use the quotient property of exponents.

00:01:36.887 --> 00:01:38.395
You have an a to the ninth

00:01:38.471 --> 00:01:43.621
Let me use a slightly different color. We have an a to the ninth.

00:01:43.621 --> 00:01:47.733
over a squared. What's that going to simplify to?

00:01:47.733 --> 00:01:49.302
Well, that's going to simplify to be the same thing,

00:01:49.318 --> 00:01:49.818
let me write this

00:01:49.818 --> 00:01:56.677
a to the ninth over a squared, the same thing as a to the nine minus two, which is equal to

00:01:56.677 --> 00:01:59.031
a to the seventh power.

00:01:59.031 --> 00:01:59.975
Now we also have

00:01:59.975 --> 00:02:02.036
and this will get a little bit interesting here

00:02:02.036 --> 00:02:05.025
We have a b squared over b to the third power

00:02:05.025 --> 00:02:07.867
So that simplifies too.

00:02:07.867 --> 00:02:10.118
So b squared over b to the third is equal to

00:02:10.118 --> 00:02:12.975
b to the two minus three power, which is equal to

00:02:12.975 --> 00:02:13.513
b to the negative one power.

00:02:13.944 --> 00:02:16.631
And we'll leave it alone like that right now.

00:02:16.631 --> 00:02:18.790
So this whole expression simplifies to

00:02:18.790 --> 00:02:27.810
It simplifies to: 5 times a to the seventh power (because this simplifies to a to the seventh)

00:02:28.533 --> 00:02:33.092
a to the seventh, times (the bs right here simplify too)

00:02:33.142 --> 00:02:36.600
b to the negative one.

00:02:36.600 --> 00:02:37.979
We could leave it like that, you know, that's pretty simple

00:02:37.979 --> 00:02:41.215
but we may not want a negative exponent there

00:02:41.215 --> 00:02:45.421
we just have to remeber that b to the negative one power is the same thing as

00:02:45.421 --> 00:02:48.779
one over b. Now if we remember that, then we can rewrite this

00:02:48.779 --> 00:02:52.036
entire expression as, the numerator will have a five

00:02:52.036 --> 00:02:54.913
and will have a to the seventh

00:02:54.913 --> 00:02:58.692
5 a to the seventh. And then the denominator will have the b.

00:02:58.723 --> 00:03:02.179
So we're multiplying this times one over b.

00:03:02.179 --> 00:03:04.523
That's the same thing as b to the negative one.

00:03:04.523 --> 99:59:59.999
And we are done!


Exponent properties 6

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:05.630
Simplify 2a squared
over 3b to the fifth,

00:00:05.630 --> 00:00:07.740
all of that to the fourth power.

00:00:07.740 --> 00:00:11.740
So here we can use the property
that if I have x over y,

00:00:11.740 --> 00:00:14.630
and I'm raising that
to the nth power,

00:00:14.630 --> 00:00:18.830
this is the same thing as
x to the nth power over y

00:00:18.830 --> 00:00:19.690
to the nth power.

00:00:19.690 --> 00:00:21.910
And we could see that
with an example real fast.

00:00:21.910 --> 00:00:26.890
If I had, let's say I had x
over y to the third power,

00:00:26.890 --> 00:00:28.380
what would this be equal to?

00:00:28.380 --> 00:00:33.770
This would be x over y
times x over y times x

00:00:33.770 --> 00:00:37.550
over y, which is going
to be equal to x times x

00:00:37.550 --> 00:00:41.890
times x over y times y times y.

00:00:41.890 --> 00:00:45.870
And this is clearly x to the
third over y to the third.

00:00:45.870 --> 00:00:48.862
So I just gave you a particular
example of why this works.

00:00:48.862 --> 00:00:50.570
Hopefully, it gives
you enough confidence

00:00:50.570 --> 00:00:53.270
or makes you feel good
about using this property.

00:00:53.270 --> 00:00:56.050
So let's use it right here
in our actual problem.

00:00:56.050 --> 00:00:58.410
So we have 2a squared
over 3b to the fifth,

00:00:58.410 --> 00:01:00.580
all of that to the fourth power.

00:01:00.580 --> 00:01:04.370
So this is going to be
the same thing as 2a

00:01:04.370 --> 00:01:05.704
squared to the fourth power.

00:01:05.704 --> 00:01:06.870
So let me write it this way.

00:01:06.870 --> 00:01:11.460
It's going to be 2a squared
to the fourth power.

00:01:11.460 --> 00:01:13.000
Let me do the fourth
power in blue.

00:01:15.750 --> 00:01:25.800
All of that over 3b to the
fifth to the fourth power.

00:01:28.802 --> 00:01:30.510
And now we can use
the property that if I

00:01:30.510 --> 00:01:32.960
take the product of
things and raise that

00:01:32.960 --> 00:01:37.120
to a power, that's the
same thing as raising each

00:01:37.120 --> 00:01:39.240
of the things in the
product to the power

00:01:39.240 --> 00:01:40.610
and then taking their product.

00:01:40.610 --> 00:01:42.193
So this is going to
be the same thing.

00:01:42.193 --> 00:01:44.190
Let me just write
the property here.

00:01:44.190 --> 00:01:49.870
So if I have x times
y to the nth power,

00:01:49.870 --> 00:01:53.390
this is the same thing
as x to the nth times

00:01:53.390 --> 00:01:55.322
y to the nth power.

00:01:55.322 --> 00:01:56.780
And you could do
a similar argument

00:01:56.780 --> 00:01:59.310
that we did right here to
see why this works out.

00:01:59.310 --> 00:02:00.640
So let's do that over here.

00:02:00.640 --> 00:02:06.020
2a squared to the fourth power
is going to be equal to 2

00:02:06.020 --> 00:02:08.390
to the fourth power--
I'll color code

00:02:08.390 --> 00:02:12.740
it-- 2 to the fourth power times
a squared to the fourth power.

00:02:17.160 --> 00:02:21.870
And then all of that over 3b to
the fifth to the fourth power.

00:02:21.870 --> 00:02:28.350
That's going to be 3 to
the fourth power times b

00:02:28.350 --> 00:02:33.940
to the fifth to
the fourth power.

00:02:36.336 --> 00:02:38.210
Now, the last interesting
thing here-- and we

00:02:38.210 --> 00:02:40.740
can evaluate what these
numbers come out to be.

00:02:40.740 --> 00:02:43.750
2 to the fourth power-- let
me just do it right now.

00:02:43.750 --> 00:02:47.480
2 to the fourth--
2 squared is 4.

00:02:47.480 --> 00:02:49.910
2 to the third power is 8.

00:02:49.910 --> 00:02:51.900
2 to the fourth power is 16.

00:02:51.900 --> 00:02:54.310
So this right here is 16.

00:02:54.310 --> 00:02:56.010
3 to the first power is 3.

00:02:56.010 --> 00:02:57.570
3 squared is 9.

00:02:57.570 --> 00:02:59.710
3 to the third power is 27.

00:02:59.710 --> 00:03:02.370
3 to the fourth power is 81.

00:03:02.370 --> 00:03:05.620
So this over here
is-- let me do it

00:03:05.620 --> 00:03:10.500
in that green
color-- this is 81.

00:03:10.500 --> 00:03:12.070
And to simplify
these terms here,

00:03:12.070 --> 00:03:14.860
we just have to remember the--I
guess we call it the power

00:03:14.860 --> 00:03:17.380
property or the power
rule of exponents.

00:03:17.380 --> 00:03:23.140
And that's just the notion
that if I have x to the n

00:03:23.140 --> 00:03:26.120
and then I raise that
to the mth power,

00:03:26.120 --> 00:03:31.676
this is the exact same thing
as x to the n times m power.

00:03:31.676 --> 00:03:33.050
So if we wanted
to simplify this,

00:03:33.050 --> 00:03:39.100
our numerator is now 16--
that's the 2 to the fourth--

00:03:39.100 --> 00:03:41.920
and then a squared and
then that the fourth power.

00:03:41.920 --> 00:03:45.460
That's going to be a
to the 2 times 4 power,

00:03:45.460 --> 00:03:48.310
or a to the eighth power.

00:03:48.310 --> 00:03:50.540
This is literally 2
times 4, so we're just

00:03:50.540 --> 00:03:52.980
multiplying 2 times 4.

00:03:52.980 --> 00:03:56.250
This 2 times this
4 to get that 8.

00:03:56.250 --> 00:04:01.130
And then in the denominator,
we have our 81 and then

00:04:01.130 --> 00:04:03.590
b to the fifth and then
that the fourth power.

00:04:03.590 --> 00:04:07.330
That's going to be b to
the 5 times 4 power or b

00:04:07.330 --> 00:04:08.710
to the 20th power.

00:04:08.710 --> 00:04:12.970
So I got 20 by
multiplying 5 times 4.

00:04:12.970 --> 00:04:14.200
And we're done.

00:04:14.200 --> 00:04:18.540
This is about as
simplified as we can get.


Exponent properties 5

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.370
Simplify 4a squared b
and this whole quantity

00:00:04.370 --> 00:00:06.487
is being raised to
the third power.

00:00:06.487 --> 00:00:08.070
Now here, we'll just
use the property.

00:00:08.070 --> 00:00:10.080
If we have the
product of things--

00:00:10.080 --> 00:00:13.500
so let's say I have a times
b-- and I'm raising that

00:00:13.500 --> 00:00:15.850
to some power n,
this is just going

00:00:15.850 --> 00:00:20.376
to be equal to a to
the n times b to the n.

00:00:20.376 --> 00:00:22.190
And to realize why
that happens, let's

00:00:22.190 --> 00:00:26.930
just try it out with
a simple example.

00:00:26.930 --> 00:00:31.194
If I have a times b
to the third power,

00:00:31.194 --> 00:00:32.610
what is that going
to be equal to?

00:00:32.610 --> 00:00:35.840
That is going to be
equal to a times b--

00:00:35.840 --> 00:00:38.580
that to the first power--
then times a times b--

00:00:38.580 --> 00:00:41.760
that's to the second
power-- times a times b--

00:00:41.760 --> 00:00:43.060
that's to the third power.

00:00:43.060 --> 00:00:44.610
That's exactly what--
we're literally

00:00:44.610 --> 00:00:47.240
taking this expression and
multiplying by itself three

00:00:47.240 --> 00:00:48.350
times.

00:00:48.350 --> 00:00:50.740
Now, we can just swap
these around because

00:00:50.740 --> 00:00:52.840
of the commutative and
the associative properties

00:00:52.840 --> 00:00:54.399
of multiplication.

00:00:54.399 --> 00:00:55.940
So if we just swap
these around, just

00:00:55.940 --> 00:00:57.980
change the order that
we're multiplying,

00:00:57.980 --> 00:01:02.820
we can rewrite this as a
times a times a times b times

00:01:02.820 --> 00:01:04.430
b times b.

00:01:04.430 --> 00:01:06.630
We have three a's
here, three b's.

00:01:06.630 --> 00:01:09.530
And so this over here is
going to be a to the third,

00:01:09.530 --> 00:01:11.010
and this right
over here is going

00:01:11.010 --> 00:01:12.800
to be equal to b to third.

00:01:12.800 --> 00:01:14.950
So we have a to the third
times b to the third.

00:01:14.950 --> 00:01:16.720
So it worked, and
so hopefully that

00:01:16.720 --> 00:01:18.980
gives you a sense of
why this property holds.

00:01:18.980 --> 00:01:22.600
Now, let's apply it
to the actual problem.

00:01:22.600 --> 00:01:25.750
So we have 4a squared
b to the third power.

00:01:25.750 --> 00:01:28.550
That means that each of these
components in the product

00:01:28.550 --> 00:01:31.420
are going to be raised
to the third power.

00:01:31.420 --> 00:01:34.890
So that means it equals
4 to the third power

00:01:34.890 --> 00:01:39.240
times a squared to
the third power times

00:01:39.240 --> 00:01:40.516
b to the third power.

00:01:40.516 --> 00:01:41.390
Let me color code it.

00:01:41.390 --> 00:01:44.120
This third power
here in magenta,

00:01:44.120 --> 00:01:46.510
that's this third
power over here.

00:01:46.510 --> 00:01:48.990
I'm just raising-- instead
of taking the product first

00:01:48.990 --> 00:01:50.910
and then raising that
to the third power,

00:01:50.910 --> 00:01:54.020
I can take each of the
terms in the product, each

00:01:54.020 --> 00:01:55.940
of the numbers in the
product, and raise those

00:01:55.940 --> 00:01:59.490
to the third power first
and then take the product.

00:01:59.490 --> 00:02:00.620
So that's what we're doing.

00:02:00.620 --> 00:02:01.911
And let's try to simplify this.

00:02:01.911 --> 00:02:05.720
So what is 4 to the third power?

00:02:05.720 --> 00:02:09.039
4 to the first is 4,
4 to the second is 16,

00:02:09.039 --> 00:02:10.979
and then we want to
multiply 16 times 4.

00:02:10.979 --> 00:02:16.200
So 4 to the third power, which
is 4 times 4 times 4, is 64.

00:02:16.200 --> 00:02:19.030
And then we have
a squared, and we

00:02:19.030 --> 00:02:20.680
are raising that
to the third power.

00:02:20.680 --> 00:02:22.380
Well, we know from
the product property

00:02:22.380 --> 00:02:26.850
of exponents, that's going to
be a to the 2 times 3 power,

00:02:26.850 --> 00:02:28.890
or a to the sixth power.

00:02:28.890 --> 00:02:30.840
So this is a to the sixth power.

00:02:30.840 --> 00:02:34.130
We got 6 from 2 times 3.

00:02:34.130 --> 00:02:37.905
And then finally, we just
have the b to the third power.

00:02:37.905 --> 00:02:39.800
Let me do that in
a different color.

00:02:39.800 --> 00:02:41.655
We just have the b
to the third power,

00:02:41.655 --> 00:02:43.250
and we'll just
write it over here.

00:02:43.250 --> 00:02:46.510
So we simplify this
expression to 64 a

00:02:46.510 --> 00:02:49.469
to the sixth b to
the third power.


Exponent properties 7

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:05.260
Simplify xy to the
negative 2 power times 2x

00:00:05.260 --> 00:00:09.270
squared over y squared
to the fourth power.

00:00:09.270 --> 00:00:11.440
So let's see what
we can do over here.

00:00:11.440 --> 00:00:14.480
So let's think about this
part of our expression first.

00:00:14.480 --> 00:00:18.790
If I have x over y
to the n-th power,

00:00:18.790 --> 00:00:23.220
we know that this is the same
thing as x to the n-th over y

00:00:23.220 --> 00:00:24.360
to the n-th.

00:00:24.360 --> 00:00:26.270
So let's apply that
right over here.

00:00:26.270 --> 00:00:29.380
So this expression, 2x squared
over y squared to the fourth,

00:00:29.380 --> 00:00:33.870
this is going to be the
same thing as 2x squared

00:00:33.870 --> 00:00:40.970
to the fourth power over y
squared to the fourth power.

00:00:40.970 --> 00:00:42.960
And of course, we
still have our xy

00:00:42.960 --> 00:00:45.590
to the negative
2 power out here.

00:00:45.590 --> 00:00:47.420
And actually, let me
simplify that as well.

00:00:47.420 --> 00:00:53.590
We know that if we have x
times y to the n-th power,

00:00:53.590 --> 00:00:56.960
this is the same thing as taking
each of the powers of each

00:00:56.960 --> 00:00:59.680
of the x and the y and
then taking the product.

00:00:59.680 --> 00:01:03.075
This is x to the n
times y to the n.

00:01:03.075 --> 00:01:04.660
And actually, if
you just expand--

00:01:04.660 --> 00:01:06.800
if you multiply
xy times itself n

00:01:06.800 --> 00:01:08.720
times you'll see why
this actually works out

00:01:08.720 --> 00:01:11.730
because you'll have x multiplied
by itself x times if you

00:01:11.730 --> 00:01:16.260
rearrange the product and y
multiplied by itself n times.

00:01:16.260 --> 00:01:17.830
So if we use that
same property here,

00:01:17.830 --> 00:01:20.310
xy to the negative
2, that's going

00:01:20.310 --> 00:01:25.560
to be the same exact thing
as x to the negative 2 times

00:01:25.560 --> 00:01:27.780
y to the negative 2.

00:01:27.780 --> 00:01:29.180
Now what can we simplify?

00:01:29.180 --> 00:01:31.900
Well, we can use this property
again right over here.

00:01:31.900 --> 00:01:34.720
We have 2x squared raised
to the fourth power.

00:01:34.720 --> 00:01:39.150
So that's going to be
the same thing as 2

00:01:39.150 --> 00:01:45.530
to the fourth power times x
squared to the fourth power.

00:01:45.530 --> 00:01:47.320
And we can do it
in the denominator.

00:01:47.320 --> 00:01:49.410
y squared to the fourth
power-- actually,

00:01:49.410 --> 00:01:51.868
we're going to use a different
property in the denominator.

00:01:51.868 --> 00:01:53.710
We just have y squared
to the fourth power.

00:01:53.710 --> 00:01:56.950
We know that if
we have x to the n

00:01:56.950 --> 00:01:59.390
and then raise that
to the m-th power,

00:01:59.390 --> 00:02:02.750
this is the same thing
as x to the n times m.

00:02:02.750 --> 00:02:04.447
So y squared and
then raising that

00:02:04.447 --> 00:02:06.030
to the fourth power
that's going to be

00:02:06.030 --> 00:02:09.270
equal to y to the
2 times 4 power,

00:02:09.270 --> 00:02:13.130
or y to the eighth power.

00:02:13.130 --> 00:02:16.620
And of course, out here we
have x to the negative 2 times

00:02:16.620 --> 00:02:18.420
y to the negative 2.

00:02:18.420 --> 00:02:21.340
Now, let's think a little bit
about what x to the negative 2

00:02:21.340 --> 00:02:23.290
and y to the negative 2 mean.

00:02:23.290 --> 00:02:27.210
x to the negative 2 just
means 1 over x squared.

00:02:27.210 --> 00:02:31.430
y to the negative 2 just
means 1 over y squared.

00:02:31.430 --> 00:02:32.960
So let's rewrite it like that.

00:02:32.960 --> 00:02:39.170
So this is the same thing
as 1 over x squared times 1

00:02:39.170 --> 00:02:40.250
over y squared.

00:02:40.250 --> 00:02:43.710
So we could just write that
as 1 over x squared y squared.

00:02:43.710 --> 00:02:45.897
That and that are
completely equivalent.

00:02:45.897 --> 00:02:47.230
So let's see what we can do now.

00:02:47.230 --> 00:02:49.522
So we have one last
thing we could-- well,

00:02:49.522 --> 00:02:50.730
we don't have one last thing.

00:02:50.730 --> 00:02:52.590
We have a few more
things to worry about.

00:02:52.590 --> 00:02:54.750
We have x squared and
then we're raising that

00:02:54.750 --> 00:02:55.730
to the fourth power.

00:02:55.730 --> 00:02:58.250
Once again, we can use
this property right here.

00:02:58.250 --> 00:03:02.100
This is going to be x
to the 2 times 4 power.

00:03:02.100 --> 00:03:05.254
So this part right over here--
I'll do it in a new color.

00:03:05.254 --> 00:03:06.670
This part right
over here is going

00:03:06.670 --> 00:03:11.790
to simplify to x to the
eighth power 2 times 4.

00:03:11.790 --> 00:03:15.380
This part right over here is
going to 2 to the fourth power.

00:03:15.380 --> 00:03:19.430
To the third is 8 so
2 to the fourth is 16.

00:03:19.430 --> 00:03:21.420
So that is 16.

00:03:21.420 --> 00:03:23.820
And so our whole expression
has now simplified to--

00:03:23.820 --> 00:03:25.272
and if we multiply
it times this,

00:03:25.272 --> 00:03:26.730
we multiply it
times 1 so I'm going

00:03:26.730 --> 00:03:28.660
to multiply these two fractions.

00:03:28.660 --> 00:03:32.020
My numerator 16 times
x to the eighth.

00:03:35.190 --> 00:03:38.140
And all of that over
I have x squared.

00:03:42.090 --> 00:03:44.217
And then I'm going to
have a y squared times a y

00:03:44.217 --> 00:03:44.800
to the eighth.

00:03:44.800 --> 00:03:46.780
What is y squared
times y to the eighth?

00:03:46.780 --> 00:03:49.790
y squared times y to the eighth.

00:03:49.790 --> 00:03:52.120
We have the same base so
we can add the exponents.

00:03:52.120 --> 00:03:53.828
We're taking the
product so it's to going

00:03:53.828 --> 00:03:57.350
to be y to the 2 plus 8,
or y to the 10th power.

00:03:57.350 --> 00:03:59.310
So y squared times
y the eighth is

00:03:59.310 --> 00:04:02.790
going to be y to the 10th power.

00:04:02.790 --> 00:04:03.950
And we're almost there.

00:04:03.950 --> 00:04:05.630
We have one more
thing we can simplify.

00:04:05.630 --> 00:04:08.750
We have x to the
eighth over x squared.

00:04:08.750 --> 00:04:11.202
And we know-- I'll just write
the general property-- if we

00:04:11.202 --> 00:04:16.750
have x to the m over
x to the n, this

00:04:16.750 --> 00:04:21.620
is equal to x to
the m minus n power.

00:04:21.620 --> 00:04:25.450
So x to the eighth over x
squared, that over there

00:04:25.450 --> 00:04:30.290
will simplify to x to
the 8 minus 2 power

00:04:30.290 --> 00:04:33.310
or x to the sixth power.

00:04:33.310 --> 00:04:40.000
And so our whole expression has
now simplified to 16 times x

00:04:40.000 --> 00:04:46.380
to the sixth all of that
over y to the 10th power.

00:04:48.940 --> 00:04:51.110
And we are done.


Multiplying in scientific notation (example)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.590
Multiply, expressing the
product in scientific notation.

00:00:04.590 --> 00:00:06.670
So let's multiply
first, and then let's

00:00:06.670 --> 00:00:09.094
try to get what we have
in scientific notation.

00:00:09.094 --> 00:00:10.510
Actually, before
we do that, let's

00:00:10.510 --> 00:00:13.480
just even remember what it means
to be in scientific notation.

00:00:13.480 --> 00:00:15.480
To be in scientific
notation-- and actually,

00:00:15.480 --> 00:00:18.750
each of these numbers right
here are in scientific notation.

00:00:18.750 --> 00:00:21.950
It's going to be
the form a times 10

00:00:21.950 --> 00:00:27.820
to some power, where a can be
greater than or equal to 1,

00:00:27.820 --> 00:00:30.670
and it is going to
be less than 10.

00:00:30.670 --> 00:00:33.227
So both of these numbers are
greater than or equal to 1,

00:00:33.227 --> 00:00:34.810
and they are less
than 10, and they're

00:00:34.810 --> 00:00:37.220
being multiplied by
some power of 10.

00:00:37.220 --> 00:00:38.940
Let's see how we
could multiply this.

00:00:38.940 --> 00:00:42.060
So this over here, this is
just the exact same thing.

00:00:42.060 --> 00:00:44.420
So if I do this in magenta,
this is the exact same thing

00:00:44.420 --> 00:00:54.410
as 9.1 times 10 to the sixth
times 3.2 times-- actually,

00:00:54.410 --> 00:00:55.620
I don't have to write it.

00:00:55.620 --> 00:00:57.600
Let me write it all
with a dot notation

00:00:57.600 --> 00:00:59.475
to make it a little bit
more straightforward.

00:01:02.095 --> 00:01:03.555
I'm doing that in magenta.

00:01:03.555 --> 00:01:12.150
This is equal to 9.1
times 10 to the sixth--

00:01:12.150 --> 00:01:17.760
let me do it in this green
color-- times 3.2 times 10

00:01:17.760 --> 00:01:19.970
to the negative 5th power.

00:01:19.970 --> 00:01:22.260
Now in multiplication,
this comes

00:01:22.260 --> 00:01:23.740
from the associative property.

00:01:23.740 --> 00:01:26.540
It essentially allows us to
remove these parentheses.

00:01:26.540 --> 00:01:28.560
It says, look, you can
multiply like that first,

00:01:28.560 --> 00:01:30.630
or you could actually
multiply these guys first.

00:01:30.630 --> 00:01:32.040
You can reassociate them.

00:01:32.040 --> 00:01:33.605
And the commutative
property tells us

00:01:33.605 --> 00:01:36.004
that we can rearrange
this thing right here.

00:01:36.004 --> 00:01:37.420
What I want to
rearrange is I want

00:01:37.420 --> 00:01:40.860
to multiply the 9.1 times
the 3.2 first and then

00:01:40.860 --> 00:01:43.382
multiply that times 10 to the
sixth times 10 the negative 5.

00:01:43.382 --> 00:01:44.840
So I'm just going
to rearrange this

00:01:44.840 --> 00:01:46.740
using the commutative property.

00:01:46.740 --> 00:01:54.410
This is the same thing
as 9.1 times 3.2,

00:01:54.410 --> 00:01:56.060
and I'm going to reassociate.

00:01:56.060 --> 00:02:00.920
So I'm going to do these
first, and then that times 10

00:02:00.920 --> 00:02:05.920
to the sixth times
10 to the negative 5.

00:02:08.440 --> 00:02:09.990
And the reason
why this is useful

00:02:09.990 --> 00:02:12.070
is that this is really
easy to multiply.

00:02:12.070 --> 00:02:15.610
We have the same
base here, base 10,

00:02:15.610 --> 00:02:18.370
and we're taking the product,
so we can add the exponents.

00:02:18.370 --> 00:02:23.239
So this part right over here,
10 to the sixth times 10

00:02:23.239 --> 00:02:24.780
to the negative 5,
that's going to be

00:02:24.780 --> 00:02:29.180
10 to the 6 minus 5 power,
or essentially just 10

00:02:29.180 --> 00:02:32.370
to the first power, which
is really just equal to 10.

00:02:32.370 --> 00:02:36.590
And that's going to be
multiplied by 9.1 times 3.2.

00:02:36.590 --> 00:02:37.760
So let me do that over here.

00:02:37.760 --> 00:02:41.767
If I have 9.1 times
3.2, so at first

00:02:41.767 --> 00:02:43.600
I'm going to ignore the
decimal, so I'm just

00:02:43.600 --> 00:02:45.690
going to treat it
like 91 times 32.

00:02:45.690 --> 00:02:47.695
So I have 2 times 1 is 2.

00:02:47.695 --> 00:02:50.020
2 times 9 is 18.

00:02:50.020 --> 00:02:52.330
I'll stick a 0 here because
I'm in the tens place

00:02:52.330 --> 00:02:55.210
now, multiplying everything
really by 30 not just by 3.

00:02:55.210 --> 00:02:57.100
That's why my zero is there.

00:02:57.100 --> 00:02:59.840
And I multiplied 3
times 1 to get 3,

00:02:59.840 --> 00:03:03.230
and then 3 times 9 is 27.

00:03:03.230 --> 00:03:05.220
And so it is 2.

00:03:05.220 --> 00:03:06.290
So I'm adding here.

00:03:06.290 --> 00:03:07.860
2 plus 0 is 2.

00:03:07.860 --> 00:03:11.980
8 plus 3 is 11, carry
or regroup that 1.

00:03:11.980 --> 00:03:14.360
1 plus 1 is 2.

00:03:14.360 --> 00:03:17.990
2 plus 7 is 9.

00:03:17.990 --> 00:03:19.760
And then I have a 2 here.

00:03:19.760 --> 00:03:22.800
So 91 times 32 is 2,912.

00:03:22.800 --> 00:03:24.740
But I didn't
multiply 91 times 32.

00:03:24.740 --> 00:03:28.110
I multiplied 9.1 times 3.2.

00:03:28.110 --> 00:03:29.850
So what I want to do
is count the number

00:03:29.850 --> 00:03:32.050
of digits I have behind
the decimal point.

00:03:32.050 --> 00:03:35.571
I have one, two digits
behind the decimal point,

00:03:35.571 --> 00:03:38.070
and so I'll have to have two
digits behind the decimal point

00:03:38.070 --> 00:03:38.690
in the answer.

00:03:38.690 --> 00:03:42.330
So one, two, I'll stick the
decimal right over there.

00:03:42.330 --> 00:03:49.550
This part right over here
comes out to be 29.12.

00:03:49.550 --> 00:03:50.890
You might feel like we're done.

00:03:50.890 --> 00:03:53.150
This kind of looks like
scientific notation.

00:03:53.150 --> 00:03:55.970
I have a number
times a power of 10.

00:03:55.970 --> 00:04:01.010
But remember, this number has to
be greater than or equal to 1--

00:04:01.010 --> 00:04:03.410
which it is-- and less than 10.

00:04:03.410 --> 00:04:05.240
But this number is
not less than 10.

00:04:05.240 --> 00:04:07.534
It's not in scientific notation.

00:04:07.534 --> 00:04:08.950
What we can do is
let's just write

00:04:08.950 --> 00:04:11.100
this number in
scientific notation,

00:04:11.100 --> 00:04:12.830
and then we can use
the power of 10 part

00:04:12.830 --> 00:04:14.490
to multiply by this power of 10.

00:04:17.540 --> 00:04:23.890
29.12, this is the
same thing as 2.912.

00:04:23.890 --> 00:04:26.270
Notice, what did I do to
go from there to there?

00:04:26.270 --> 00:04:28.410
I just moved the
decimal to the left.

00:04:28.410 --> 00:04:29.980
Or another way to
think about it,

00:04:29.980 --> 00:04:34.060
if I wanted to go from here to
there, what could I do to this?

00:04:34.060 --> 00:04:35.660
Well, I would multiply it by 10.

00:04:35.660 --> 00:04:37.800
If I multiplied it by 10,
I would move the decimal

00:04:37.800 --> 00:04:38.380
to the right.

00:04:38.380 --> 00:04:41.590
It would go from 2.9 to 29.

00:04:41.590 --> 00:04:43.120
So if I want to
write this value,

00:04:43.120 --> 00:04:46.900
this is just this times 10.

00:04:46.900 --> 00:04:52.360
So 29.12 is the same
thing as 2.912 times 10.

00:04:52.360 --> 00:04:54.250
Now, this is in
scientific notation,

00:04:54.250 --> 00:04:55.750
but that's just this part.

00:04:55.750 --> 00:04:58.600
And I still have to
multiply it by another 10,

00:04:58.600 --> 00:05:01.510
so times another 10.

00:05:01.510 --> 00:05:09.322
To finish up this problem, we
get 2.912 times 10 times 10,

00:05:09.322 --> 00:05:11.030
or 10 to the first
times 10 to the first.

00:05:11.030 --> 00:05:12.130
Well, what's that?

00:05:12.130 --> 00:05:15.040
Well, that's going to be
this part right over here.

00:05:15.040 --> 00:05:16.740
That's just 10 squared.

00:05:16.740 --> 00:05:23.170
So it's 2.912 times 10 to the
second power, and we are done.


Express a small number in scientific notation (example)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:11.012
Express 0.0000000003457
in scientific notation.

00:00:11.012 --> 00:00:12.470
So let's just remind
ourselves what

00:00:12.470 --> 00:00:15.250
it means to be in
scientific notation.

00:00:15.250 --> 00:00:20.850
Scientific notation will be some
number times some power of 10

00:00:20.850 --> 00:00:24.330
where this number right here--
let me write it this way.

00:00:24.330 --> 00:00:27.340
It's going to be greater
than or equal to 1,

00:00:27.340 --> 00:00:30.790
and it's going to
be less than 10.

00:00:30.790 --> 00:00:32.689
So over here, what
we want to put here

00:00:32.689 --> 00:00:34.920
is what that leading
number is going to be.

00:00:34.920 --> 00:00:36.420
And in general,
you're going to look

00:00:36.420 --> 00:00:37.994
for the first non-zero digit.

00:00:37.994 --> 00:00:39.410
And this is the
number that you're

00:00:39.410 --> 00:00:40.743
going to want to start off with.

00:00:40.743 --> 00:00:43.790
This is the only number you're
going to want to put ahead of

00:00:43.790 --> 00:00:46.490
or I guess to the left
of the decimal point.

00:00:46.490 --> 00:00:53.200
So we could write
3.457, and it's

00:00:53.200 --> 00:00:56.397
going to be multiplied
by 10 to something.

00:00:56.397 --> 00:00:57.980
Now let's think about
what we're going

00:00:57.980 --> 00:00:59.380
to have to multiply it by.

00:00:59.380 --> 00:01:06.110
To go from 3.457 to this
very, very small number,

00:01:06.110 --> 00:01:08.230
from 3.457, to get
to this, you have

00:01:08.230 --> 00:01:10.300
to move the decimal
to the left a bunch.

00:01:10.300 --> 00:01:12.960
You have to add a bunch of
zeroes to the left of the 3.

00:01:12.960 --> 00:01:17.720
You have to keep moving the
decimal over to the left.

00:01:17.720 --> 00:01:19.540
To do that, we're
essentially making

00:01:19.540 --> 00:01:22.236
the number much
much, much smaller.

00:01:22.236 --> 00:01:23.610
So we're not going
to multiply it

00:01:23.610 --> 00:01:25.790
by a positive exponent of 10.

00:01:25.790 --> 00:01:29.360
We're going to multiply it
times a negative exponent of 10.

00:01:29.360 --> 00:01:31.370
The equivalent is
you're dividing

00:01:31.370 --> 00:01:33.827
by a positive exponent of 10.

00:01:33.827 --> 00:01:35.410
And so the best way
to think about it,

00:01:35.410 --> 00:01:40.130
when you move an
exponent one to the left,

00:01:40.130 --> 00:01:44.240
you're dividing by 10, which
is equivalent to multiplying

00:01:44.240 --> 00:01:48.310
by 10 to the negative 1 power.

00:01:48.310 --> 00:01:50.530
Let me give you example here.

00:01:50.530 --> 00:01:55.140
So if I have 1 times 10 is
clearly just equal to 10.

00:01:55.140 --> 00:01:57.750
1 times 10 to the
negative 1, that's

00:01:57.750 --> 00:02:03.140
equal to 1 times 1/10,
which is equal to 1/10.

00:02:03.140 --> 00:02:05.710
1 times-- and let me actually
write a decimal, which is equal

00:02:05.710 --> 00:02:09.930
to 0-- let me actually-- I
skipped a step right there.

00:02:09.930 --> 00:02:13.760
Let me add 1 times 10 to the 0,
so we have something natural.

00:02:13.760 --> 00:02:15.600
So this is one times
10 to the first.

00:02:15.600 --> 00:02:18.600
One times 10 to the 0
is equal to 1 times 1,

00:02:18.600 --> 00:02:20.290
which is equal to 1.

00:02:20.290 --> 00:02:26.140
1 times 10 to the negative
1 is equal to 1/10,

00:02:26.140 --> 00:02:29.160
which is equal to 0.1.

00:02:29.160 --> 00:02:32.715
If I do 1 times 10
to the negative 2,

00:02:32.715 --> 00:02:36.820
10 to the negative 2 is 1
over 10 squared or 1/100.

00:02:36.820 --> 00:02:41.900
So this is going to be
1/100, which is 0.01.

00:02:41.900 --> 00:02:42.820
What's happening here?

00:02:42.820 --> 00:02:46.640
When I raise it to
a negative 1 power,

00:02:46.640 --> 00:02:48.240
I've essentially
moved the decimal

00:02:48.240 --> 00:02:51.334
from to the right of the
1 to the left of the 1.

00:02:51.334 --> 00:02:52.750
I've moved it from
there to there.

00:02:52.750 --> 00:02:54.390
When I raise it
to the negative 2,

00:02:54.390 --> 00:02:56.680
I moved it two over to the left.

00:02:56.680 --> 00:03:00.390
So how many times are we
going to have to move it over

00:03:00.390 --> 00:03:04.730
to the left to get this
number right over here?

00:03:04.730 --> 00:03:06.940
So let's think about
how many zeroes we have.

00:03:06.940 --> 00:03:10.010
So we have to move it one time
just to get in front of the 3.

00:03:10.010 --> 00:03:12.040
And then we have to
move it that many more

00:03:12.040 --> 00:03:15.310
times to get all of the zeroes
in there so that we have

00:03:15.310 --> 00:03:17.604
to move it one
time to get the 3.

00:03:17.604 --> 00:03:19.020
So if we started
here, we're going

00:03:19.020 --> 00:03:28.130
to move 1, 2, 3, 4, 5,
6, 7, 8, 9, 10 times.

00:03:28.130 --> 00:03:32.931
So this is going to be 3.457
times 10 to the negative 10

00:03:32.931 --> 00:03:33.430
power.

00:03:33.430 --> 00:03:34.388
Let me just rewrite it.

00:03:34.388 --> 00:03:39.764
So 3.457 times 10 to
the negative 10 power.

00:03:39.764 --> 00:03:41.180
So in general,
what you want to do

00:03:41.180 --> 00:03:44.450
is you want to find the
first non-zero number here.

00:03:44.450 --> 00:03:47.060
Remember, you want a number
here that's between 1 and 10.

00:03:47.060 --> 00:03:49.910
And it can be equal to 1, but
it has to be less than 10.

00:03:49.910 --> 00:03:53.320
3.457 definitely fits that bill.

00:03:53.320 --> 00:03:55.760
It's between 1 and 10.

00:03:55.760 --> 00:03:57.840
And then you just want
to count the leading

00:03:57.840 --> 00:03:59.636
zeroes between the
decimal and that number

00:03:59.636 --> 00:04:01.510
and include the number
because that tells you

00:04:01.510 --> 00:04:04.470
how many times you have
to shift the decimal over

00:04:04.470 --> 00:04:06.166
to actually get
this number up here.

00:04:06.166 --> 00:04:08.040
And so we have to shift
this decimal 10 times

00:04:08.040 --> 00:04:11.840
to the left to get
this thing up here.


Speed of light and distance from sun (scientific notation word problem)

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:05.750
The speed of light is 3
times 10 to the eighth meters

00:00:05.750 --> 00:00:06.600
per second.

00:00:06.600 --> 00:00:08.510
So as you can tell,
light is very fast,

00:00:08.510 --> 00:00:11.150
3 times 10 to the eighth
meters per second.

00:00:11.150 --> 00:00:16.100
If it takes 5 times 10 to the
second power seconds for light

00:00:16.100 --> 00:00:18.740
to travel from the
sun to the earth--

00:00:18.740 --> 00:00:20.380
let's think about
that a little bit.

00:00:20.380 --> 00:00:24.630
5 times 10 to the second,
that's 500 seconds.

00:00:24.630 --> 00:00:27.510
You have 60 seconds in
a minute, so 8 minutes

00:00:27.510 --> 00:00:30.720
would be 480 seconds.

00:00:30.720 --> 00:00:34.310
So 500 seconds would be
about 8 minutes, 20 seconds.

00:00:34.310 --> 00:00:36.230
It takes 8 minutes,
20 seconds for light

00:00:36.230 --> 00:00:39.100
to travel from the
sun to the earth.

00:00:39.100 --> 00:00:45.070
What is the distance, in meters,
between the sun and the earth?

00:00:45.070 --> 00:00:46.880
They're giving us a rate.

00:00:46.880 --> 00:00:48.470
They're giving us a speed.

00:00:48.470 --> 00:00:49.880
They're giving us a time.

00:00:49.880 --> 00:00:52.070
And they want to
find a distance.

00:00:52.070 --> 00:00:55.050
This goes straight back
to the standard distance

00:00:55.050 --> 00:00:58.760
is equal to rate times time.

00:00:58.760 --> 00:01:00.800
So they give us the rate.

00:01:00.800 --> 00:01:04.269
The rate is 3 times 10 to
the eighth meters per second.

00:01:10.050 --> 00:01:12.840
That right there is the rate.

00:01:12.840 --> 00:01:14.780
They give us the time.

00:01:14.780 --> 00:01:17.305
The time is 5 times 10
to the second seconds.

00:01:24.740 --> 00:01:26.770
I'll just use that with a S.

00:01:26.770 --> 00:01:27.750
How many meters?

00:01:27.750 --> 00:01:29.090
So what is the distance?

00:01:31.790 --> 00:01:36.320
And so we can just move these
around from the commutative

00:01:36.320 --> 00:01:40.692
and the associative
properties of multiplication.

00:01:40.692 --> 00:01:42.400
And actually, you can
multiply the units.

00:01:42.400 --> 00:01:44.025
That's called
dimensional analysis.

00:01:44.025 --> 00:01:46.150
When you multiply the units,
you kind of treat them

00:01:46.150 --> 00:01:47.110
like variables.

00:01:47.110 --> 00:01:49.447
You should get the right
dimensions for distance.

00:01:49.447 --> 00:01:51.030
So let's just rearrange
these numbers.

00:01:51.030 --> 00:01:56.510
This is equal to 3
times 5-- I'm just

00:01:56.510 --> 00:02:00.665
commuting and reassociating
these numbers and this product,

00:02:00.665 --> 00:02:04.640
because we're just multiplying
everything-- 3 times 5 times

00:02:04.640 --> 00:02:10.439
10 to the eighth times
10 to the second.

00:02:14.250 --> 00:02:19.690
And then we're going to
have meters per second times

00:02:19.690 --> 00:02:21.794
seconds.

00:02:21.794 --> 00:02:23.460
And if you treated
these like variables,

00:02:23.460 --> 00:02:26.990
these seconds would cancel out
with that seconds right there,

00:02:26.990 --> 00:02:30.100
and you would just be left with
the unit meters, which is good,

00:02:30.100 --> 00:02:33.882
because we want a
distance in just meters.

00:02:33.882 --> 00:02:34.840
How does this simplify?

00:02:34.840 --> 00:02:38.560
This gives us 3 times 5 is 15.

00:02:38.560 --> 00:02:41.520
15 times 10 to the
eighth times 10 squared.

00:02:41.520 --> 00:02:42.450
We have the same base.

00:02:42.450 --> 00:02:45.450
We're taking the product,
so we can add the exponents.

00:02:45.450 --> 00:02:49.840
This is going to be 10 to
the 8 plus 2 power, or 10

00:02:49.840 --> 00:02:51.440
to the 10th power.

00:02:51.440 --> 00:02:54.270
Now you might be tempted
to say that we're done,

00:02:54.270 --> 00:02:56.150
that we have this in
scientific notation.

00:02:56.150 --> 00:02:58.590
But remember, in
scientific notation

00:02:58.590 --> 00:03:01.570
this number here has
to be greater than

00:03:01.570 --> 00:03:04.010
or equal to 1 and less than 10.

00:03:04.010 --> 00:03:06.590
This clearly is
not less than 10.

00:03:06.590 --> 00:03:08.090
So how do we rewrite this?

00:03:08.090 --> 00:03:11.350
We can write 15 as 1.5.

00:03:11.350 --> 00:03:13.900
This clearly is greater
than 1 and less than 10.

00:03:13.900 --> 00:03:20.430
And to get from 1.5 to 15,
you have to multiply by 10.

00:03:20.430 --> 00:03:23.610
One way to think about
it is 15 is 15.0,

00:03:23.610 --> 00:03:24.955
and so you have a decimal here.

00:03:33.100 --> 00:03:36.870
If we're moving the decimal
one to the left to make it 1.5,

00:03:36.870 --> 00:03:39.090
that's essentially
dividing by 10.

00:03:39.090 --> 00:03:41.805
Moving the decimal to the left
means you're dividing by 10.

00:03:41.805 --> 00:03:43.930
If we don't want to change
the value of the number,

00:03:43.930 --> 00:03:47.530
we need to divide by 10
and then multiply by 10.

00:03:47.530 --> 00:03:50.220
So this and that
are the same number.

00:03:50.220 --> 00:03:52.960
Now 15 is 1.5 times
10, and then we

00:03:52.960 --> 00:04:01.640
have to multiply that
times 10 to the 10th power,

00:04:01.640 --> 00:04:02.820
this right over here.

00:04:02.820 --> 00:04:05.400
10 is really just 10
to the first power.

00:04:05.400 --> 00:04:06.790
So we can just
add the exponents.

00:04:06.790 --> 00:04:08.620
Same base, taking the product.

00:04:08.620 --> 00:04:14.350
This is equal to 1.5 times
10 to the 1 plus 10 power,

00:04:14.350 --> 00:04:16.990
or 10 to the 11th power.

00:04:16.990 --> 00:04:17.980
And we are done.

00:04:17.980 --> 00:04:20.089
This is a huge distance.

00:04:23.394 --> 00:04:24.560
It's very hard to visualize.

00:04:24.560 --> 00:04:27.010
But anyway, hopefully
you enjoyed that.


Example 1: Simplifying polynomials

WEBVTT
Kind: captions
Language: en

00:00:01.133 --> 00:00:11.000
simply 3x squared minus 8x plus 7 plus 2x to the third minus x squared plus eight x minus 3

00:00:11.000 --> 00:00:14.800
so when we simplify this we're essentially going to add up like terms

00:00:14.800 --> 00:00:19.133
and just as a reminder we can only add or subtract like terms or simplify like terms

00:00:19.133 --> 00:00:24.533
and just a reminder and what I mean by that if I had an x squared to an x squared

00:00:24.533 --> 00:00:28.467
these are like terms they're both x terms raised to the same power

00:00:28.467 --> 00:00:30.067
the same degree

00:00:30.067 --> 00:00:35.800
so if I have one x squared and another x squared well then I have 2x squared - this is 2x squared

00:00:35.800 --> 00:00:41.733
If I have an x to the third - let's say I have 3x to thirds plus another 4x to the thirds

00:00:41.733 --> 00:00:45.400
Well that means I have 7x to the thirds - 7x to the thirds

00:00:45.400 --> 00:00:53.333
I can't take an x squared and add it to an x to the third I cannot simplify this in any way, so this you cannot simplify

00:00:53.333 --> 00:00:59.333
cannot simply - these are not like terms just because they both have Xs

00:00:59.333 --> 00:01:03.400
The Xs are not to - they are not to the same degree

00:01:03.400 --> 00:01:06.733
With that in mind let's look at the Xs to the same degree

00:01:06.733 --> 00:01:08.133
Let's start with the highest degree

00:01:08.133 --> 00:01:12.533
So the highest degree or the highest exponent on an x here - is actually this x to the third here

00:01:12.533 --> 00:01:17.200
but it looks like the only one it's the only place where we're raising x to the third power

00:01:17.200 --> 00:01:21.800
so that can't be merged or added or subtracted to anything else

00:01:21.800 --> 00:01:23.467
so let's just write that down.

00:01:23.467 --> 00:01:25.333
so we have 2x to the third

00:01:25.333 --> 00:01:31.200
and let's look at the x squared terms. We have 3x squared over there

00:01:31.200 --> 00:01:37.133
and we have a minus or we can view as a negative x squared over there

00:01:37.133 --> 00:01:43.000
so if we want to simplify we can add these two terms - we can add- so let me just write it down

00:01:43.000 --> 00:01:47.800
we can add 3x squared to negative x squared

00:01:47.800 --> 00:01:50.933
so I'm just rearranging it really right now

00:01:50.933 --> 00:01:54.400
I'm putting the like terms next to each other so it'll be easy to simplify

00:01:54.400 --> 00:01:58.200
now let's just worry about the x to the first terms or just the x terms

00:01:58.200 --> 00:02:00.867
you have a negative 8x term right over here

00:02:00.867 --> 00:02:03.000
so let me write it over here

00:02:03.000 --> 00:02:09.000
negative 8x and then you have a positive 8x term right over here, so let me write that down

00:02:09.000 --> 00:02:11.067
so positive 8x

00:02:11.067 --> 00:02:13.733
and then finally let's look at the constant terms

00:02:13.733 --> 00:02:16.733
you can view those as times x to the zeroth power

00:02:16.733 --> 00:02:21.400
and the constant terms are - you have a positive 7 over here - so plus 7

00:02:21.400 --> 00:02:24.533
and then you have a negative 3 over here

00:02:24.533 --> 00:02:27.067
you have a negative 3

00:02:27.067 --> 00:02:31.133
so all I'll I've done is I've really just used the communative property of addition

00:02:31.133 --> 00:02:36.667
to just change the order - or addition and subtraction - to change the order in which I'm doing this

00:02:36.667 --> 00:02:40.467
I've just rearranged the things so the like terms are next to each other

00:02:40.467 --> 00:02:41.800
but now we can simplify

00:02:41.800 --> 00:02:45.333
so we have 2x to the third - nothing to simplify that with

00:02:45.333 --> 00:02:50.733
but then if we subtract - if we have - let me do that in the same blue color

00:02:50.733 --> 00:02:55.200
if we have 3x squared and from that we're taking away an x squared

00:02:55.200 --> 00:02:57.733
well, we're only gonna have 2x squared left

00:02:57.733 --> 00:03:01.067
so that's gonna be plus 2x squared

00:03:01.067 --> 00:03:04.667
and then over here if we have negative 8 Xs

00:03:04.667 --> 00:03:08.400
and then we add 8 Xs to it or you can actually swap these around

00:03:08.400 --> 00:03:15.067
you could view it as you are subtracting 8 x from positive 8x

00:03:15.067 --> 00:03:16.733
we'll those are just going to cancel out

00:03:16.733 --> 00:03:22.067
so it's just going to be zero - I could just write plus zero here, but that'd just be redundant it wouldn't change the value

00:03:22.067 --> 00:03:27.067
and then finally I have a plus 7 minus 3 well that is just clearly

00:03:27.067 --> 00:03:30.533
7 minus 3 is 4 so I have plus 4

00:03:30.533 --> 00:03:32.667
And we're done! We've simplified it!

00:03:32.667 --> 99:59:59.999
2x to the third plus 2 x squared plus 4


The parts of polynomial expressions

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:04.430
In the following polynomial,
identify the terms along

00:00:04.430 --> 00:00:07.640
with the coefficient and
exponent of each term.

00:00:07.640 --> 00:00:10.590
So the terms are just
the things being added up

00:00:10.590 --> 00:00:12.470
in this polynomial.

00:00:12.470 --> 00:00:17.180
So the terms here-- let
me write the terms here.

00:00:17.180 --> 00:00:20.760
The first term is 3x squared.

00:00:20.760 --> 00:00:25.290
The second term it's being
added to negative 8x.

00:00:25.290 --> 00:00:27.490
You might say, hey
wait, isn't it minus 8x?

00:00:27.490 --> 00:00:28.989
And you could just
view that as it's

00:00:28.989 --> 00:00:30.470
being added to negative 8x.

00:00:30.470 --> 00:00:32.940
So negative 8x is
the second term.

00:00:32.940 --> 00:00:35.410
And then the third
term here is 7.

00:00:35.410 --> 00:00:37.050
It's called a polynomial.

00:00:37.050 --> 00:00:39.460
Poly, it has many terms.

00:00:39.460 --> 00:00:41.610
Or you could view each
term as a monomial,

00:00:41.610 --> 00:00:45.390
as a polynomial with
only one term in it.

00:00:45.390 --> 00:00:47.007
So those are the terms.

00:00:47.007 --> 00:00:48.590
Now let's think about
the coefficients

00:00:48.590 --> 00:00:49.830
of each of the terms.

00:00:49.830 --> 00:00:53.720
The coefficient is what's
multiplying the power of x

00:00:53.720 --> 00:00:56.960
or what's multiplying in
the x part of the term.

00:00:56.960 --> 00:00:59.900
So over here, the x
part is x squared.

00:00:59.900 --> 00:01:01.815
That's being multiplied by 3.

00:01:01.815 --> 00:01:05.180
So 3 is the coefficient
on the first term.

00:01:05.180 --> 00:01:08.460
On the second term, we have
negative 8 multiplying x.

00:01:08.460 --> 00:01:10.710
And we want to be clear, the
coefficient isn't just 8.

00:01:10.710 --> 00:01:12.490
It's a negative 8.

00:01:12.490 --> 00:01:15.600
It's negative 8
that's multiplying x.

00:01:15.600 --> 00:01:17.700
So that's the coefficient
right over here.

00:01:17.700 --> 00:01:19.270
And here you might
say, hey wait,

00:01:19.270 --> 00:01:21.580
nothing is multiplying x here.

00:01:21.580 --> 00:01:22.430
I just have a 7.

00:01:22.430 --> 00:01:23.540
There is no x.

00:01:23.540 --> 00:01:25.460
7 isn't being multiplied by x.

00:01:25.460 --> 00:01:30.560
But you can think of this as 7
being multiplied by x to the 0

00:01:30.560 --> 00:01:34.310
because we know that x to the
zeroth power is equal to 1.

00:01:34.310 --> 00:01:37.890
So we would even call
this constant, the 7,

00:01:37.890 --> 00:01:41.490
this would be the
coefficient on 7x to the 0.

00:01:41.490 --> 00:01:47.660
So you could view
this as a coefficient.

00:01:47.660 --> 00:01:49.105
So this is also a coefficient.

00:01:49.105 --> 00:01:53.393
So let me make it clear, these
three things are coefficients.

00:01:59.130 --> 00:02:01.030
Now the last part, they
want us to identify

00:02:01.030 --> 00:02:03.420
the exponent of each term.

00:02:03.420 --> 00:02:06.180
So the exponent of
this first term is 2.

00:02:06.180 --> 00:02:08.120
It's being raised
to the second power.

00:02:08.120 --> 00:02:11.160
The exponent of the second
term, remember, negative 8x, x

00:02:11.160 --> 00:02:13.810
is the same thing as
x to the first power.

00:02:13.810 --> 00:02:15.920
So the exponent here is 1.

00:02:15.920 --> 00:02:18.520
And then on this last
term, we already said,

00:02:18.520 --> 00:02:21.230
7 is the same thing
as 7x to the 0.

00:02:21.230 --> 00:02:26.560
So the exponent here on the
constant term on 7 is 0.

00:02:26.560 --> 00:02:33.780
So these things right over
here, those are our exponents.

00:02:33.780 --> 00:02:36.190
And we are done.


Evaluating a polynomial at a given value

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.360
Evaluate 3x squared
minus 8x plus 7

00:00:04.360 --> 00:00:07.420
when x is equal to negative 2.

00:00:07.420 --> 00:00:10.830
So to evaluate this expression
when x is equal to negative 2,

00:00:10.830 --> 00:00:12.700
everywhere that we
see an x, we just

00:00:12.700 --> 00:00:14.800
have to substitute
it with a negative 2.

00:00:14.800 --> 00:00:17.950
So it would be 3 times,
instead of x squared,

00:00:17.950 --> 00:00:22.010
it would be 3 times negative
2 squared minus 8 times--

00:00:22.010 --> 00:00:27.300
instead of 8 times x, 8
times negative 2 plus 7.

00:00:27.300 --> 00:00:28.850
And so what does this give us?

00:00:28.850 --> 00:00:32.150
So we have 3 times
negative 2 squared.

00:00:32.150 --> 00:00:33.855
Well, that's just positive 4.

00:00:33.855 --> 00:00:36.520
A negative times a
negative is a positive.

00:00:36.520 --> 00:00:44.170
Minus 8 times negative
2 is negative 16 plus 7.

00:00:44.170 --> 00:00:47.760
So 3 times 4 is 12.

00:00:47.760 --> 00:00:49.940
From that, you're
subtracting negative 16.

00:00:49.940 --> 00:00:51.880
Subtracting negative
16 is the same thing

00:00:51.880 --> 00:00:53.650
as adding positive 16.

00:00:53.650 --> 00:00:57.700
So it's 12 plus 16 plus 7.

00:00:57.700 --> 00:01:01.940
And so this is equal
to 12 plus 16 is 28.

00:01:01.940 --> 00:01:07.620
And then if you add
7 to 28, you get 35.


Example 4: Adding and subtracting polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:05.990
Add, and they give us 5x squared
plus 8x minus 3 plus 2x squared

00:00:05.990 --> 00:00:08.420
minus 7x plus 13x.

00:00:08.420 --> 00:00:11.292
So we can really view this as
just adding two polynomials.

00:00:11.292 --> 00:00:12.750
And actually, the
second polynomial

00:00:12.750 --> 00:00:15.300
here can be simplified
right off the bat.

00:00:15.300 --> 00:00:18.310
We have two like terms here.

00:00:18.310 --> 00:00:19.534
It's not the 2x squared.

00:00:19.534 --> 00:00:21.700
There's no other 2x squared
term in this polynomial.

00:00:21.700 --> 00:00:23.770
But you have a negative 7x term.

00:00:23.770 --> 00:00:25.690
And then you have a plus 13x.

00:00:25.690 --> 00:00:28.030
So we could actually
add these two terms.

00:00:28.030 --> 00:00:30.310
What is the negative
7 of something

00:00:30.310 --> 00:00:31.700
plus 13 of that something?

00:00:31.700 --> 00:00:34.880
Or another way to view
it, what's 13x minus 7x?

00:00:34.880 --> 00:00:37.577
13 of something minus
7 of that something?

00:00:37.577 --> 00:00:39.660
Well, that's just going
to be 6 of that something.

00:00:39.660 --> 00:00:41.610
So that is just going to be 6x.

00:00:41.610 --> 00:00:44.140
And then you have
your 2x squared.

00:00:44.140 --> 00:00:46.920
And you have your 2x
squared right over here.

00:00:46.920 --> 00:00:47.950
Let me write it.

00:00:47.950 --> 00:00:49.874
So 2x squared.

00:00:49.874 --> 00:00:51.290
So this polynomial
right over here

00:00:51.290 --> 00:00:54.210
simplifies to 2x
squared plus 6x.

00:00:54.210 --> 00:00:56.380
And then we want to add it
to this polynomial right

00:00:56.380 --> 00:00:56.880
over here.

00:00:56.880 --> 00:00:58.790
And what we can do
is, what I like to do

00:00:58.790 --> 00:01:02.010
is just rewrite this polynomial
under this polynomial.

00:01:02.010 --> 00:01:04.450
And I'm going to align
the similar terms

00:01:04.450 --> 00:01:06.325
or the like terms,
the terms that

00:01:06.325 --> 00:01:09.030
have the same variable
raised to the same power.

00:01:09.030 --> 00:01:12.340
So we have x raised
to the second power.

00:01:12.340 --> 00:01:14.910
So let's put the 5x
squared over here.

00:01:14.910 --> 00:01:17.900
We have an x raised
to the first power.

00:01:17.900 --> 00:01:19.660
We have an x raised
to the first power.

00:01:19.660 --> 00:01:22.320
So let's put the 8x over here.

00:01:22.320 --> 00:01:24.740
And then we don't
have a constant term

00:01:24.740 --> 00:01:26.230
in this yellow polynomial.

00:01:26.230 --> 00:01:27.750
We don't have anything
raised to the zeroth power,

00:01:27.750 --> 00:01:28.541
any constant terms.

00:01:28.541 --> 00:01:29.650
We do here.

00:01:29.650 --> 00:01:31.690
So let's just put a minus 3.

00:01:31.690 --> 00:01:33.920
And then we could add
these two things up.

00:01:33.920 --> 00:01:36.700
We have 2x squared
plus 5x squared.

00:01:36.700 --> 00:01:40.470
That is 2 of something
plus 5 of something

00:01:40.470 --> 00:01:42.440
is 7 of that something.

00:01:42.440 --> 00:01:44.740
So it's going to be 7x squared.

00:01:44.740 --> 00:01:46.610
And then to that,
if I 6 of something

00:01:46.610 --> 00:01:48.910
and I add 8 of that
something to it,

00:01:48.910 --> 00:01:51.010
well, I'm going to have
14 of that something.

00:01:51.010 --> 00:01:53.620
If I have 6x's and I give
you 8 more x's, you're going

00:01:53.620 --> 00:01:54.931
to 14x's.

00:01:54.931 --> 00:01:56.180
And then we have nothing here.

00:01:56.180 --> 00:01:59.060
And so if we just add
nothing to negative 3,

00:01:59.060 --> 00:02:01.970
you're just going
to get a negative 3.

00:02:01.970 --> 00:02:04.950
So this simplifies to, or when
we added these two polynomials,

00:02:04.950 --> 00:02:09.390
we get 7x squared
plus 14x minus 3.


Example 3: Subtracting polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:05.510
Simplify 16x plus 14 minus
the entire expression 3x

00:00:05.510 --> 00:00:08.340
squared plus x minus 9.

00:00:08.340 --> 00:00:10.510
So when you subtract
an entire expression,

00:00:10.510 --> 00:00:16.329
this is the exact same
thing as having 16x plus 14.

00:00:16.329 --> 00:00:19.280
And then you're adding the
opposite of this whole thing.

00:00:19.280 --> 00:00:24.680
Or you're adding
negative 1 times 3x

00:00:24.680 --> 00:00:27.485
squared plus x minus 9.

00:00:27.485 --> 00:00:29.110
Or another way to
think about it is you

00:00:29.110 --> 00:00:32.450
can distribute this negative
sign along all of those terms.

00:00:32.450 --> 00:00:34.760
That's essentially what
we're about to do here.

00:00:34.760 --> 00:00:37.450
We're just adding the
negative of this entire thing.

00:00:37.450 --> 00:00:39.061
We're adding the opposite of it.

00:00:39.061 --> 00:00:41.060
So this first part-- I'm
not going to change it.

00:00:41.060 --> 00:00:43.710
That's still just 16x plus 14.

00:00:43.710 --> 00:00:46.310
But now I'm going to distribute
the negative sign here.

00:00:46.310 --> 00:00:52.780
So negative 1 times 3x squared
is negative 3x squared.

00:00:52.780 --> 00:00:56.690
Negative 1 times
positive x is negative

00:00:56.690 --> 00:00:59.170
x because that's positive 1x.

00:00:59.170 --> 00:01:02.174
Negative 1 times
negative 9-- remember,

00:01:02.174 --> 00:01:04.340
you have to consider this
negative right over there.

00:01:04.340 --> 00:01:05.960
That is part of the term.

00:01:05.960 --> 00:01:08.470
Negative 1 times
negative 9 is positive 9.

00:01:08.470 --> 00:01:11.070
Negative times a
negative is a positive.

00:01:11.070 --> 00:01:14.240
So then we have positive 9.

00:01:14.240 --> 00:01:16.255
And now we just have
to combine like terms.

00:01:16.255 --> 00:01:17.880
So what's our highest
degree term here?

00:01:17.880 --> 00:01:19.254
I like to write
it in that order.

00:01:19.254 --> 00:01:22.487
We have only one x squared
term, second-degree term.

00:01:22.487 --> 00:01:23.570
We only have one of those.

00:01:23.570 --> 00:01:26.610
So let me write it over
here-- negative 3x squared.

00:01:26.610 --> 00:01:29.050
And then what do we have in
terms of first-degree terms,

00:01:29.050 --> 00:01:30.970
of just an x, x to
the first power?

00:01:30.970 --> 00:01:33.070
Well, we have a 16x.

00:01:33.070 --> 00:01:36.710
And then from that, we're going
to subtract an x, subtract 1x.

00:01:36.710 --> 00:01:40.820
So 16x minus 1x is 15x.

00:01:40.820 --> 00:01:43.470
If you have 16 of something and
you subtract 1 of them away,

00:01:43.470 --> 00:01:45.460
you're going to have
15 of that something.

00:01:45.460 --> 00:01:48.370
And then finally, you have 14.

00:01:48.370 --> 00:01:51.866
You could view that as 14
times x to the 0 or just 14.

00:01:51.866 --> 00:01:55.380
14 plus 9-- they're
both constant terms,

00:01:55.380 --> 00:01:57.530
or they're both being
multiplied by x to the 0.

00:01:57.530 --> 00:02:01.325
14 plus 9 is 23.

00:02:01.325 --> 00:02:01.950
And we're done.

00:02:01.950 --> 00:02:06.071
Negative 3x squared
plus 15x plus 23.


Example 3: Multiplying a monomial by a polynomial

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.320
Multiply negative 4x squared
by the whole expression

00:00:04.320 --> 00:00:07.780
3x squared plus 25x minus 7.

00:00:07.780 --> 00:00:10.016
So if you multiply anything
times a whole expression,

00:00:10.016 --> 00:00:11.890
you really just use the
distributive property

00:00:11.890 --> 00:00:14.440
to multiply each term
of the expression

00:00:14.440 --> 00:00:16.131
by the negative 4x squared.

00:00:16.131 --> 00:00:17.630
So we're going to
have to distribute

00:00:17.630 --> 00:00:22.400
this negative 4x squared over
every term in the expression.

00:00:22.400 --> 00:00:24.530
So first, we could
start with negative 4x

00:00:24.530 --> 00:00:26.715
squared times 3x squared.

00:00:26.715 --> 00:00:27.590
So we can write that.

00:00:27.590 --> 00:00:32.990
We're going to have negative
4x squared times 3x squared.

00:00:32.990 --> 00:00:38.320
And to that, we're going to add
negative 4x squared times 25x.

00:00:41.890 --> 00:00:46.440
And to that, we're going
to add negative 4x squared

00:00:46.440 --> 00:00:48.465
times negative 7.

00:00:51.640 --> 00:00:53.890
So let's just simplify
this a little bit.

00:00:53.890 --> 00:00:56.306
Now, we can obviously
swap the order.

00:00:56.306 --> 00:00:57.930
We're just multiplying
negative 4 times

00:00:57.930 --> 00:00:59.469
x squared times 3
times x squared.

00:00:59.469 --> 00:01:01.010
And actually, I'll
do out every step.

00:01:01.010 --> 00:01:03.140
Eventually, you can do
some of this in your head.

00:01:03.140 --> 00:01:07.430
This is the exact same thing
as negative 4 times 3 times

00:01:07.430 --> 00:01:11.147
x squared times x squared.

00:01:11.147 --> 00:01:12.230
And what is that equal to?

00:01:12.230 --> 00:01:15.820
Well, negative 4 times
3 is negative 12.

00:01:15.820 --> 00:01:18.180
And x squared times x
squared-- same base.

00:01:18.180 --> 00:01:19.430
We're taking the product.

00:01:19.430 --> 00:01:21.740
That's going to be
x to the fourth.

00:01:21.740 --> 00:01:24.930
So this right here is
negative 12x to the fourth.

00:01:24.930 --> 00:01:28.340
Now let's think about
this term over here.

00:01:28.340 --> 00:01:30.540
This is the same thing
as-- and of course,

00:01:30.540 --> 00:01:32.880
we have this plus out here.

00:01:32.880 --> 00:01:35.230
And then this part right
here is the exact same thing

00:01:35.230 --> 00:01:44.120
as 25 times negative 4
times x squared times x.

00:01:44.120 --> 00:01:46.150
So let's just multiply
the numbers out here.

00:01:46.150 --> 00:01:47.520
These were the coefficients.

00:01:47.520 --> 00:01:50.447
25 times negative
4 is negative 100.

00:01:50.447 --> 00:01:52.280
So it'll plus negative
100, or we could just

00:01:52.280 --> 00:01:54.820
say it's minus 100.

00:01:54.820 --> 00:01:56.610
And then we have
x squared times x,

00:01:56.610 --> 00:01:59.770
or x squared times x
to the first power.

00:01:59.770 --> 00:02:02.070
Same base-- we can
add the exponents.

00:02:02.070 --> 00:02:03.690
2 plus 1 is 3.

00:02:03.690 --> 00:02:08.020
So this is negative
100x to the third power.

00:02:08.020 --> 00:02:10.320
And then let's look at
this last term over here.

00:02:10.320 --> 00:02:11.871
We have negative 4x squared.

00:02:11.871 --> 00:02:13.370
So this is going
to be plus-- that's

00:02:13.370 --> 00:02:16.030
this plus right over here.

00:02:16.030 --> 00:02:17.790
We have negative 4.

00:02:17.790 --> 00:02:20.390
We can multiply that
times negative 7.

00:02:20.390 --> 00:02:22.730
And then multiply
that times x squared.

00:02:22.730 --> 00:02:25.240
I'm just changing the order
in which we multiply it.

00:02:25.240 --> 00:02:30.130
So negative 4 times
negative 7 is positive 28.

00:02:30.130 --> 00:02:35.010
And then I'm going to multiply
that times the x squared.

00:02:35.010 --> 00:02:37.450
There's no simplification
to do, no like terms.

00:02:37.450 --> 00:02:39.200
These are different powers of x.

00:02:39.200 --> 00:02:41.876
So we are done.