Subtracting and simplifying radicals

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.320
We're asked to subtract all
of this craziness over here.

00:00:04.320 --> 00:00:05.430
And it looks daunting.

00:00:05.430 --> 00:00:08.029
But if we really just
focus, it actually

00:00:08.029 --> 00:00:10.560
should be pretty straightforward
to subtract and simplify

00:00:10.560 --> 00:00:11.490
this thing.

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Because right from the get-go,
I have 4 times the fourth root

00:00:15.220 --> 00:00:16.915
of 81x to the fifth.

00:00:16.915 --> 00:00:18.290
And from that, I
want to subtract

00:00:18.290 --> 00:00:23.180
2 times the fourth root
of 81x to the fifth.

00:00:23.180 --> 00:00:26.180
And so you really can just say,
look, I have four of something.

00:00:26.180 --> 00:00:28.530
And then the something
I'll just circle in yellow.

00:00:28.530 --> 00:00:30.440
I have four of this--
it could be lemons.

00:00:30.440 --> 00:00:32.110
I have four of
these things, and I

00:00:32.110 --> 00:00:33.870
want to subtract
two of these things.

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These are the exact same things.

00:00:35.770 --> 00:00:39.980
They're the fourth root
of 81x to the fifth.

00:00:39.980 --> 00:00:43.920
So if I have four lemons and
I want to subtract two lemons,

00:00:43.920 --> 00:00:45.780
I'm going to have
two lemons left over.

00:00:45.780 --> 00:00:47.780
Or if I have four of this
thing, and I take away

00:00:47.780 --> 00:00:49.470
two of this thing,
I'm going to have

00:00:49.470 --> 00:00:51.830
two of these things left over.

00:00:51.830 --> 00:00:56.000
So these terms right over
here simplify to 2 times

00:00:56.000 --> 00:01:00.520
the fourth root of
81x to the fifth.

00:01:00.520 --> 00:01:02.793
And I got this 2 just by
subtracting the coefficients.

00:01:02.793 --> 00:01:04.459
Four of something
minus two of something

00:01:04.459 --> 00:01:07.300
is equal to two
of that something.

00:01:07.300 --> 00:01:08.790
And then that, of
course, we still

00:01:08.790 --> 00:01:13.040
have this minus the regular
principal square root

00:01:13.040 --> 00:01:16.200
of x to the third.

00:01:16.200 --> 00:01:19.680
Now I want to try to
simplify what's inside

00:01:19.680 --> 00:01:22.482
of these under the radical
sign so that we can,

00:01:22.482 --> 00:01:24.440
in this example, actually
take the fourth root,

00:01:24.440 --> 00:01:29.240
and over here actually take,
maybe, a principal square root.

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So first of all, let's
see if 81 is something

00:01:32.447 --> 00:01:34.030
to the fourth power,
or at least could

00:01:34.030 --> 00:01:35.490
be factored into
something that is

00:01:35.490 --> 00:01:37.760
a something to the fourth power.

00:01:37.760 --> 00:01:41.560
So 81, if we do prime
factorization, is 3 times 27.

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27 is 3 times 9.

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And 9 is 3 times 3.

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So 81 is exactly 3
times 3 times 3 times 3.

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So 81 actually is 3 to
the fourth power, which

00:01:52.485 --> 00:01:53.860
is convenient
because we're going

00:01:53.860 --> 00:01:56.320
to be taking the
fourth root of that.

00:01:56.320 --> 00:01:59.457
And then x to the fifth
we can write as a product.

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Let me write it over here
so it doesn't get messy.

00:02:01.540 --> 00:02:04.080
So I'm going to write what's
under the radical as 3

00:02:04.080 --> 00:02:10.389
to the fourth power times x
to the fourth power times x.

00:02:10.389 --> 00:02:13.790
x to the fourth times x
is x to the fifth power.

00:02:13.790 --> 00:02:16.170
And I'm taking the fourth
root of all of this.

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And taking the fourth root
of all of this-- that's

00:02:18.500 --> 00:02:21.375
the same thing as taking
the fourth root of this,

00:02:21.375 --> 00:02:23.470
as taking the
fourth root of this.

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And I'm going to
want to skip steps.

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So I'm taking the fourth root
of all of it right over there.

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And of course, I
have a 2 out front.

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And then x to the third could
be written as x squared times x.

00:02:36.350 --> 00:02:38.920
So it's minus the
principal square root

00:02:38.920 --> 00:02:41.307
of x squared times x.

00:02:41.307 --> 00:02:43.640
And I broke it up like this
because this right over here

00:02:43.640 --> 00:02:45.820
is a perfect square.

00:02:45.820 --> 00:02:48.550
Now, how can we simplify
this a little bit?

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And you're probably getting
used to the pattern.

00:02:50.550 --> 00:02:53.410
This is the same thing as the
fourth root of 3 to the fourth,

00:02:53.410 --> 00:02:55.350
times the fourth root
of x to the fourth,

00:02:55.350 --> 00:02:58.300
times the fourth root of x.

00:02:58.300 --> 00:03:00.380
So let's just skip
straight to that.

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So what is the fourth root?

00:03:02.939 --> 00:03:04.855
I could write it-- let
me write it explicitly,

00:03:04.855 --> 00:03:06.938
although you wouldn't have
to necessarily do this.

00:03:06.938 --> 00:03:12.570
This is the same thing as the
fourth root of 3 to the fourth,

00:03:12.570 --> 00:03:16.080
times the fourth root
of x to the fourth,

00:03:16.080 --> 00:03:21.370
times the fourth root of x.

00:03:21.370 --> 00:03:24.510
And 2 is being multiplied
times all of that.

00:03:24.510 --> 00:03:27.570
And then this over here is
minus the principal square root

00:03:27.570 --> 00:03:32.160
of x squared, times the
principal square root of x.

00:03:32.160 --> 00:03:34.380
And so if we try to
simplify it, the fourth root

00:03:34.380 --> 00:03:38.080
of 3 to the fourth
power is just 3.

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So we get a 3 there.

00:03:40.070 --> 00:03:42.650
The fourth root of x
to the fourth power

00:03:42.650 --> 00:03:44.110
is just going to be x.

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Actually, I just
reminded myself,

00:03:50.044 --> 00:03:51.210
we have to be careful there.

00:03:51.210 --> 00:03:54.440
It is not just x, because
what if x is negative?

00:03:54.440 --> 00:03:56.500
If x is negative, then
x to the fourth power

00:03:56.500 --> 00:03:58.150
is going to be a positive value.

00:03:58.150 --> 00:03:59.410
And when you take the
fourth-- remember,

00:03:59.410 --> 00:04:00.868
this is the fourth
principal root--

00:04:00.868 --> 00:04:03.260
you're going to get the
positive version of x.

00:04:03.260 --> 00:04:06.200
Or really, you're going to
get the absolute value of x.

00:04:06.200 --> 00:04:09.080
So here, you're
going to be getting

00:04:09.080 --> 00:04:13.460
the absolute value of x.

00:04:13.460 --> 00:04:15.590
Although, well, you
could make an argument

00:04:15.590 --> 00:04:19.170
that x needs to be positive
if this thing is going

00:04:19.170 --> 00:04:21.095
to be well-defined
in the real numbers,

00:04:21.095 --> 00:04:23.470
because then what's under the
radical has to be positive.

00:04:23.470 --> 00:04:25.760
But let's just go with
this for right now.

00:04:25.760 --> 00:04:28.860
And then we have the
fourth root of x.

00:04:28.860 --> 00:04:30.750
And then over here, the
principal square root

00:04:30.750 --> 00:04:34.670
of x squared, by
the same logic, is

00:04:34.670 --> 00:04:36.310
going to be the
absolute value of x.

00:04:36.310 --> 00:04:38.630
And then this is just the
principal square root of x.

00:04:38.630 --> 00:04:40.020
So let's multiply
everything out.

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We have 2 times 3 times
the absolute value of x.

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So 2 times 3 is 6,
times the absolute value

00:04:48.040 --> 00:04:56.400
of x, times the principal
fourth root of x, I should say,

00:04:56.400 --> 00:05:00.860
minus we took out the
absolute value of x, times

00:05:00.860 --> 00:05:02.430
the principal root of x.

00:05:02.430 --> 00:05:04.360
And we can't do any
more subtracting.

00:05:04.360 --> 00:05:07.140
Just because you have to
realize this is a fourth root.

00:05:07.140 --> 00:05:10.650
This is a regular square
root, principal square root.

00:05:10.650 --> 00:05:12.510
If these were the
same root, then maybe

00:05:12.510 --> 00:05:14.910
we could simplify this
a little bit more.

00:05:14.910 --> 00:05:16.380
And so then we are all done.

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And we have fully simplified it.

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And if you make the
assumption that this

00:05:20.390 --> 00:05:22.380
is defined for real numbers.

00:05:22.380 --> 00:05:24.210
So that the domain
over here, what

00:05:24.210 --> 00:05:27.150
has to be under these
radicals, has to be positive,

00:05:27.150 --> 00:05:28.757
actually, in every
one of these cases.

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And if they need to
be positive, we're

00:05:30.340 --> 00:05:32.298
not going to be dealing
with imaginary numbers.

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All of these need
to be positive.

00:05:33.740 --> 00:05:36.250
Their domains are x has to be
greater than or equal to 0,

00:05:36.250 --> 00:05:38.333
then you could assume that
the absolute value of x

00:05:38.333 --> 00:05:39.330
is the same as x.

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But I'll just stick
it right here.

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If you restrict the
domain, you could

00:05:42.490 --> 00:05:45.190
get rid of the
absolute value signs.


Adding and simplifying radicals

WEBVTT
Kind: captions
Language: en

00:00:01.133 --> 00:00:08.467
We're asked to add and simplify and we have the principle root of two x squared plus four times the principle

00:00:08.467 --> 00:00:14.933
root of eight plus three times the principle root two x squared plus the principle root of eight

00:00:14.933 --> 00:00:17.733
so we can do a little bit of adding, we can actually simplify

00:00:17.733 --> 00:00:19.800
first and then add or we can add first and then simplify

00:00:19.800 --> 00:00:22.400
but it looks like we can already add so lets try and do that

00:00:22.400 --> 00:00:30.133
so here, right over here, I have a principle root of two x squared and over here I have three principle

00:00:30.133 --> 00:00:35.867
roots of two x squared, well if I have one of something here and I have three of something here and i need

00:00:35.867 --> 00:00:41.800
to add them together I can put a one co-efficient out here to make it clear this is one of this thing

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and I have three of these things but if I have one of this thing and three more of these things and I add them together I am

00:00:45.733 --> 00:00:56.400
going to have four of those things, so this is four times the principle root of two x squared

00:00:56.400 --> 00:01:01.867
and that confuses a little bit, imagine that the whole principle root of two x squared was some variable

00:01:01.867 --> 00:01:08.867
lets say this whole thing was "a" and lets say that this whole thing was "a" as well, because its the same thing, you'd have one "a"

00:01:08.867 --> 00:01:14.067
plus three "a"'s which will give you four "a"'s, in this case "a" is all of this business right over here

00:01:14.067 --> 00:01:19.333
so we added those terms, and then we wanted to think about we have four principle roots of "a" and we

00:01:19.333 --> 00:01:24.800
have one more principle roots of "a", so same idea you have four of these things I am circling in magenta

00:01:24.800 --> 00:01:30.533
and you have one more of these things that I am circling in magenta, that one co-efficient is implicit

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so if I have four of something plus one more of something it becomes five of that something

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so plus plus five times the square root, plus five times the square root of eight

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and now we'll see if we can simplify this anymore, we have four of something and we have five of something

00:01:49.867 --> 00:01:54.933
else, so you can't just add these two things together, but maybe we can simplify this a little bit

00:01:54.933 --> 00:02:00.800
so we know that the principle root of two x squared, this is the same thing as, so let me write the four

00:02:00.800 --> 00:02:05.400
out front, so we have the four, and the principle root of two x squared is the same thing as the principle

00:02:05.400 --> 00:02:11.333
root of two times the principle root of x squared so I just rewrote this part over here

00:02:11.333 --> 00:02:18.867
and then we have plus five times, now eight can be written as a product of a perfect square and a

00:02:18.867 --> 00:02:26.000
not so perfect square, eight can be written as four times two, so lets write it that way

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so if we view this whole, this is the principle root, the square root of four times two, we can re-write this as the

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five times the square root of four, or the principle root of four times the principle root of two and

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what can we simplify here? well we know what the principle root of x squared is, it is the positive square

00:02:44.533 --> 00:02:52.400
root of x squared, so it is not just x, you might be tempted to say it is x but since we know it is

00:02:52.400 --> 00:02:57.467
the positive square root we have to say it is the absolute value of x, because what if x was negative?

00:02:57.467 --> 00:03:01.867
if was x was negative, you'd have , lets say it was negative three, you'd have negative three squared,

00:03:01.867 --> 00:03:08.467
you'd have a positive nine, and so the principle root of a positive nine is going to be a positive three

00:03:08.467 --> 00:03:12.733
and so it wouldn't just be x, it wouldnt be negative three, it would be positive three, so you have to

00:03:12.733 --> 00:03:17.600
take the absolute value, and the other thing that is a perfect square is the four right here, its principle

00:03:17.600 --> 00:03:24.400
root is two, its principle square root i should say is two, so now you have, if we just change the order we

00:03:24.400 --> 00:03:29.533
are multiplying right here, you have four, four times the absolute value

00:03:29.533 --> 00:03:36.467
of x, four times the absolute value of x, times the square root of two, times the square root of two,

00:03:36.467 --> 00:03:45.133
I want to do that in that same yellow color, times the square root of two, plus plus we have five times

00:03:45.133 --> 00:03:50.600
two, which is ten, right, this whole thing is simplified to two, so we have plus ten

00:03:50.600 --> 00:03:57.400
square roots of two, now we could call it a day, and say we are all done adding and simplifying

00:03:57.400 --> 00:04:02.000
or you could add a little bit more depending on how you wanna view it, because over here you have

00:04:02.000 --> 00:04:09.267
four times the absolute value of x square roots of two, and here you have ten square roots of two

00:04:09.267 --> 00:04:15.067
so you have four absolute value of x of something, and you have ten of that same something, you could

00:04:15.067 --> 00:04:19.067
add them up, or another way to think about it is, you could factor out a square root of two

00:04:19.067 --> 00:04:30.200
either one of those works, so you get four times the absolute value of x, plus ten plus ten times times

00:04:30.200 --> 00:04:37.400
the principle square root of two, so depending on whether you view this of this more simplified, one

00:04:37.400 --> 99:59:59.999
of those two will will satisfy you


Simplifying cube root expressions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.800
Multiply and simplify 5
times the cube root of 2x

00:00:03.800 --> 00:00:08.920
squared times 3 times the
cube root of 4x to the fourth.

00:00:08.920 --> 00:00:11.250
So the two things that
pop out of my brain right

00:00:11.250 --> 00:00:13.380
here is that we can change
the order a little bit

00:00:13.380 --> 00:00:15.474
because multiplication is
both commutative-- well,

00:00:15.474 --> 00:00:16.890
the commutative
property allows us

00:00:16.890 --> 00:00:18.520
to switch the order
for multiplication.

00:00:18.520 --> 00:00:20.061
And so we can get
the constant terms.

00:00:20.061 --> 00:00:21.494
We can multiply
the 5 times the 3.

00:00:21.494 --> 00:00:23.160
And then the other
two things that we're

00:00:23.160 --> 00:00:25.540
multiplying-- they're
both the cube root, which

00:00:25.540 --> 00:00:28.330
is the same thing as taking
something to the 1/3 power.

00:00:28.330 --> 00:00:31.340
So the cube root of
x-- this is exactly

00:00:31.340 --> 00:00:34.002
the same thing as
raising x to the 1/3.

00:00:34.002 --> 00:00:34.710
So let's do that.

00:00:34.710 --> 00:00:36.834
Let's switch the order and
let's rewrite these cube

00:00:36.834 --> 00:00:39.820
roots as raising it
to the 1/3 power.

00:00:39.820 --> 00:00:43.780
So I have the 5 and the 3.

00:00:43.780 --> 00:00:46.580
So that's going to be 5 times 3.

00:00:46.580 --> 00:00:51.120
And then we have the cube root
of-- do that in a new color.

00:00:51.120 --> 00:00:54.820
Then we have the cube
root of 2x squared.

00:00:54.820 --> 00:00:59.890
So this I can rewrite as 2x
squared to the 1/3 power.

00:00:59.890 --> 00:01:03.630
And then I have the cube
root of 4x to the fourth.

00:01:03.630 --> 00:01:08.110
So that's the same thing as 4x
to the fourth to the 1/3 power.

00:01:08.110 --> 00:01:10.170
And now we know from
our exponent properties,

00:01:10.170 --> 00:01:12.940
if we have two things that are
both raised to the same power

00:01:12.940 --> 00:01:14.670
and then we take
their product, we

00:01:14.670 --> 00:01:16.600
could just take their
product first and then

00:01:16.600 --> 00:01:17.650
raise it to the power.

00:01:17.650 --> 00:01:22.740
So if I have a to the x
power times b to the x power,

00:01:22.740 --> 00:01:27.080
this is the same thing as
a times b to the x power.

00:01:27.080 --> 00:01:29.040
So we can simplify this
part of the expression

00:01:29.040 --> 00:01:36.340
right over here as 2x squared
times 4x to the fourth

00:01:36.340 --> 00:01:39.310
to the 1/3 power.

00:01:39.310 --> 00:01:42.960
And of course, 5 times 3 is 15.

00:01:42.960 --> 00:01:46.030
And if we simplify what's in
the expression right over here--

00:01:46.030 --> 00:01:48.350
once again, it's commutative,
so we can swap the order.

00:01:48.350 --> 00:01:51.830
And it's associative, and so
we can swap the groupings.

00:01:51.830 --> 00:01:54.210
How we group them doesn't
matter because it's

00:01:54.210 --> 00:01:55.680
all multiplication here.

00:01:55.680 --> 00:01:59.720
This is 2 times 4, which
is 6, times x squared

00:01:59.720 --> 00:02:02.490
times x to the fourth. x
squared times x to the fourth

00:02:02.490 --> 00:02:05.260
is x to the sixth power.

00:02:05.260 --> 00:02:08.840
And it's all of that
to the 1/3 power.

00:02:08.840 --> 00:02:11.420
And then that is times
that-- oh, sorry.

00:02:11.420 --> 00:02:12.055
Not 6.

00:02:12.055 --> 00:02:13.850
2 times 4 is 8.

00:02:13.850 --> 00:02:15.390
What am I doing?

00:02:15.390 --> 00:02:17.190
2 times 4 is 8.

00:02:21.070 --> 00:02:24.930
x squared times x to the
fourth is x to the sixth.

00:02:24.930 --> 00:02:27.800
I think my brain was adding the
exponents and wrote the 6 down.

00:02:27.800 --> 00:02:30.654
Of course, 2 times
4 is 8, not 6.

00:02:30.654 --> 00:02:31.695
But we add the exponents.

00:02:31.695 --> 00:02:33.800
They have the same base.

00:02:33.800 --> 00:02:36.620
x squared times x to the
fourth is x to the sixth.

00:02:36.620 --> 00:02:39.650
And we're going to raise
that to the 1/3 power.

00:02:39.650 --> 00:02:42.484
And then all of
that is times 15.

00:02:42.484 --> 00:02:44.650
And then we essentially can
use this property again.

00:02:44.650 --> 00:02:46.380
Actually, not that property.

00:02:46.380 --> 00:02:49.740
We know that this--
well, actually,

00:02:49.740 --> 00:02:51.220
yeah, exactly this
property again.

00:02:51.220 --> 00:02:53.420
We have something
multiplied to a power.

00:02:53.420 --> 00:02:55.340
This is the exact same thing.

00:02:55.340 --> 00:02:58.820
This is the exact same
thing as 8 to the 1/3 power

00:02:58.820 --> 00:03:03.220
times x to the sixth
to the 1/3 power.

00:03:03.220 --> 00:03:09.990
And then all of that is
being multiplied by 15.

00:03:09.990 --> 00:03:11.560
And so 8 to the
1/3 power-- that's

00:03:11.560 --> 00:03:13.790
the same thing as
the cube root of 8.

00:03:13.790 --> 00:03:17.150
You might recognize that
8 is 2 times 2 times 2.

00:03:17.150 --> 00:03:20.120
So 8 to the 1/3 power is 2.

00:03:20.120 --> 00:03:21.550
8 is 2 to the third.

00:03:21.550 --> 00:03:24.110
So 2 the third to the
1/3 is 2 to the first.

00:03:24.110 --> 00:03:26.460
2 times 2 times 2 is 8.

00:03:26.460 --> 00:03:28.422
And x to the sixth
to the 1/3 we know

00:03:28.422 --> 00:03:29.630
from our exponent properties.

00:03:29.630 --> 00:03:33.020
That's the same thing as x
to the sixth times 1/3 power,

00:03:33.020 --> 00:03:35.825
or x to the 6
divided by 3 power.

00:03:35.825 --> 00:03:38.900
Or 6 divided by 3
is 2, or x squared.

00:03:38.900 --> 00:03:40.780
So that is just x squared.

00:03:40.780 --> 00:03:44.350
So you have 15 times
2, which gives us 30.

00:03:44.350 --> 00:03:46.470
So that's these terms
right over here.

00:03:46.470 --> 00:03:48.860
And then you have this
term right over here.

00:03:48.860 --> 00:03:51.106
I want to do that in
a different color.

00:03:51.106 --> 00:03:52.480
And then you have
this term right

00:03:52.480 --> 00:03:54.630
over-- that's not
a different color.

00:03:54.630 --> 00:03:58.690
You have this term right
over here is x squared,

00:03:58.690 --> 00:03:59.530
and you are done.

00:03:59.530 --> 00:04:01.540
And there's a bunch of
ways you could do it.

00:04:01.540 --> 00:04:04.180
You might not decide to
use exponent notation.

00:04:04.180 --> 00:04:07.070
You could say, look,
this is a cube root.

00:04:07.070 --> 00:04:09.617
I can then take the cube root
of the product of both of them.

00:04:09.617 --> 00:04:11.200
So you don't have
to write a 1/3 here.

00:04:11.200 --> 00:04:12.616
You could just
write the cube root

00:04:12.616 --> 00:04:14.060
of this whole thing over here.

00:04:14.060 --> 00:04:15.710
And then, depending on how
you want to group it and all

00:04:15.710 --> 00:04:17.710
the rest, you could do
this in different orders.

00:04:17.710 --> 00:04:21.140
As long as you get the
right exponent properties,

00:04:21.140 --> 00:04:24.940
you should get to
this same answer.


Multiply and simplify a radical expression 2

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.550
We're asked to
divide and simplify.

00:00:03.550 --> 00:00:05.590
And we have one
radical expression

00:00:05.590 --> 00:00:07.350
over another radical expression.

00:00:07.350 --> 00:00:09.700
The key to simplify
this is to realize

00:00:09.700 --> 00:00:14.850
if I have the principal root of
x over the principal root of y,

00:00:14.850 --> 00:00:19.697
this is the same thing as the
principal root of x over y.

00:00:19.697 --> 00:00:22.030
And it really just comes out
of the exponent properties.

00:00:22.030 --> 00:00:24.611
If I have two things that
I take to some power--

00:00:24.611 --> 00:00:27.110
and taking the principal root
is the same thing as taking it

00:00:27.110 --> 00:00:30.170
to the 1/2 power-- if
I'm raising each of them

00:00:30.170 --> 00:00:32.580
to some power and
then dividing, that's

00:00:32.580 --> 00:00:34.370
the same thing as
dividing first and then

00:00:34.370 --> 00:00:37.050
raising them to that power.

00:00:37.050 --> 00:00:38.844
So let's apply that over here.

00:00:38.844 --> 00:00:40.260
This expression
over here is going

00:00:40.260 --> 00:00:43.906
to be the same thing
as the principal root--

00:00:43.906 --> 00:00:45.280
it's hard to write
a radical sign

00:00:45.280 --> 00:00:54.700
that big-- the principal root
of 60x squared y over 48x.

00:00:54.700 --> 00:00:58.480
And then we can first look
at the coefficients of each

00:00:58.480 --> 00:01:01.390
of these expressions and
try to simplify that.

00:01:01.390 --> 00:01:05.224
Both the numerator and the
denominator is divisible by 12.

00:01:05.224 --> 00:01:07.706
60 divided by 12 is 5.

00:01:07.706 --> 00:01:10.850
48 divided by 12 is 4.

00:01:10.850 --> 00:01:13.776
Both the numerator and the
denominator are divisible by x.

00:01:13.776 --> 00:01:17.755
x squared divided
by x is just x.

00:01:17.755 --> 00:01:19.810
x divided by x is 1.

00:01:19.810 --> 00:01:21.310
Anything we divide
the numerator by,

00:01:21.310 --> 00:01:23.080
we have to divide
the denominator by.

00:01:23.080 --> 00:01:25.680
And that's all we have left.

00:01:25.680 --> 00:01:28.860
So if we wanted
to simplify this,

00:01:28.860 --> 00:01:34.640
this is equal to the--
make a radical sign--

00:01:34.640 --> 00:01:36.087
and then we have 5/4.

00:01:40.944 --> 00:01:43.360
And actually, we can write it
in a slightly different way,

00:01:43.360 --> 00:01:45.060
but I'll write it
this way-- 5/4.

00:01:45.060 --> 00:01:47.650
And we have nothing left in the
denominator other than that 4.

00:01:47.650 --> 00:01:50.010
And in the numerator, we
have an x and we have a y.

00:01:54.340 --> 00:01:57.817
And now we could leave
it just like that,

00:01:57.817 --> 00:02:00.400
but we might want to take more
things out of the radical sign.

00:02:00.400 --> 00:02:02.450
And so one possibility
that you can do

00:02:02.450 --> 00:02:05.540
is you could say that this is
really the same thing as-- this

00:02:05.540 --> 00:02:12.320
is equal to 1/4 times 5xy, all
of that under the radical sign.

00:02:12.320 --> 00:02:14.860
And this is the same
thing as the square root

00:02:14.860 --> 00:02:17.300
of or the principal
root of 1/4 times

00:02:17.300 --> 00:02:21.220
the principal root of 5xy.

00:02:21.220 --> 00:02:23.890
And the square root of
1/4, if you think about it,

00:02:23.890 --> 00:02:25.360
that's just 1/2 times 1/2.

00:02:25.360 --> 00:02:26.985
Or another way you
could think about it

00:02:26.985 --> 00:02:30.989
is that this right here
is the same thing as-- so

00:02:30.989 --> 00:02:32.530
you could just say,
hey, this is 1/2.

00:02:32.530 --> 00:02:34.390
1/2 times 1/2 is 1/4.

00:02:34.390 --> 00:02:36.140
Or if you don't realize
it's 1/2, you say,

00:02:36.140 --> 00:02:38.264
hey, this is the same thing
as the square root of 1

00:02:38.264 --> 00:02:42.250
over the square root of 4,
and the square root of 1 is 1

00:02:42.250 --> 00:02:47.520
and the principal root of 4 is
2, so you get 1/2 once again.

00:02:47.520 --> 00:02:50.110
And so if you simplify
this right here to 1/2,

00:02:50.110 --> 00:02:54.090
then the whole thing can
simplify to 1/2 times

00:02:54.090 --> 00:02:56.280
the principal root--
I'll just write it

00:02:56.280 --> 00:03:00.937
all in orange-- times the
principal root of 5xy.

00:03:00.937 --> 00:03:02.770
And there's nothing
else that you can really

00:03:02.770 --> 00:03:04.186
take out of the
radical sign here.

00:03:04.186 --> 00:03:06.750
Nothing else here
is a perfect square.


NPR Story on KA Los Altos Pilots (June 2011)

WEBVTT
Kind: captions
Language: en

00:00:00.463 --> 00:00:02.874
Robert: From NPR News this
is All Things Considered.

00:00:02.874 --> 00:00:03.789
I'm Robert Siegel.

00:00:03.789 --> 00:00:05.464
Michele: And I'm Michele Norris

00:00:05.464 --> 00:00:07.965
A lot of struggling math
students have found comfort

00:00:07.965 --> 00:00:10.852
in the sound of this man's voice.

00:00:10.852 --> 00:00:11.961
Sal: Welcome to the presentation

00:00:11.961 --> 00:00:14.213
on using the quadratic equation.

00:00:14.213 --> 00:00:16.460
Michele: Salman Kahn
started making math videos

00:00:16.460 --> 00:00:17.789
to help tutor his cousin,

00:00:17.789 --> 00:00:20.467
but they became so popular he quit his job

00:00:20.467 --> 00:00:23.240
with a hedge fund to
work on them full time.

00:00:23.240 --> 00:00:26.381
Now his online Khan
Academy attracts countless

00:00:26.381 --> 00:00:27.872
teachers and students.

00:00:27.872 --> 00:00:30.654
NPR's Larry Abramson
reports on the first effort

00:00:30.654 --> 00:00:35.152
to brings Khan's approach
into the classroom.

00:00:35.152 --> 00:00:37.739
Larry: It is shocking.
Three dissolute students

00:00:37.739 --> 00:00:40.651
in Northern California
are using school time

00:00:40.651 --> 00:00:42.351
to play a computer game.

00:00:42.351 --> 00:00:43.641
What's it called?

00:00:43.641 --> 00:00:48.375
Student: Trigonometry
Challenge. The one half. What?

00:00:48.375 --> 00:00:49.958
Student: 2X. Oh, yeah. Okay.

00:00:49.958 --> 00:00:52.552
Larry: Fifth graders
Michael Ahn, Lucas Nuin,

00:00:52.552 --> 00:00:54.382
and Reese Tumrie are actually hip deep

00:00:54.382 --> 00:00:57.469
in the Trigonometry
Challenge at Khan Academy.

00:00:57.469 --> 00:00:58.891
They're gobbling up math problems

00:00:58.891 --> 00:01:00.874
like a plate of Doritos.

00:01:00.874 --> 00:01:02.888
This school is part of a pilot program

00:01:02.888 --> 00:01:04.790
at Santa Rita Elementary School

00:01:04.790 --> 00:01:06.711
in Los Altos, California.

00:01:06.711 --> 00:01:07.956
They're making the Khan Academy

00:01:07.956 --> 00:01:10.470
an integral part of the math curriculum.

00:01:10.470 --> 00:01:13.376
Teacher Kami Thordarson
says for half an hour a day

00:01:13.376 --> 00:01:15.870
Khan Academy dunks these students into

00:01:15.870 --> 00:01:18.152
a completely self-paced world.

00:01:18.152 --> 00:01:19.839
Kami: They're all in different places.

00:01:19.839 --> 00:01:23.456
Some of them are working on
calculus and high school math.

00:01:23.456 --> 00:01:25.039
Some of them are working on multiplication

00:01:25.039 --> 00:01:27.780
of decimals, and that's okay.

00:01:27.780 --> 00:01:29.940
Larry: It's okay because
students who need help

00:01:29.940 --> 00:01:31.776
can look up the appropriate video,

00:01:31.776 --> 00:01:33.772
and listen to the academy's explanation

00:01:33.772 --> 00:01:37.364
of concepts like direct
and inverse variation.

00:01:37.364 --> 00:01:40.122
Sal: I'll do direct variation
on the left over here,

00:01:40.122 --> 00:01:42.639
and I'll do inverse variation,

00:01:42.639 --> 00:01:45.604
or two variables that vary inversely

00:01:45.604 --> 00:01:48.123
on the right hand side over here.

00:01:48.123 --> 00:01:50.339
Larry: Salman Kahn scribbles
on the video screen

00:01:50.339 --> 00:01:51.833
and guides students along

00:01:51.833 --> 00:01:55.663
with his signature approach,
patient and unthreatening.

00:01:55.663 --> 00:01:58.533
Kami Thordarson says kids
are comfortable turning

00:01:58.533 --> 00:01:59.828
to Kahn for help.

00:01:59.828 --> 00:02:01.245
Kami: It's kind of a private thing.

00:02:01.245 --> 00:02:02.243
It's not like you have to raise your hand

00:02:02.243 --> 00:02:03.341
in front of the whole class and say,

00:02:03.341 --> 00:02:06.036
"I don't get this." You can go not get it

00:02:06.036 --> 00:02:08.082
on your own and nobody
needs to know about it.

00:02:08.082 --> 00:02:11.080
Larry: There's nothing flashy
about the videos or the exercises,

00:02:11.080 --> 00:02:14.074
but somehow Khan has
clearly hit a sweet spot

00:02:14.074 --> 00:02:16.658
that makes this the most
popular part of the day.

00:02:16.658 --> 00:02:19.435
Student: You square that. That's 36,

00:02:19.435 --> 00:02:22.735
and you square that. That's 9.

00:02:22.735 --> 00:02:24.833
Larry: Kami Thordarson
says during Khan time

00:02:24.833 --> 00:02:27.657
her class looks more like
a busy engineering office

00:02:27.657 --> 00:02:29.333
than a traditional classroom.

00:02:29.333 --> 00:02:31.846
Kid's collaborate. The
teacher facilitates,

00:02:31.846 --> 00:02:33.847
and she doesn't have to lecture.

00:02:33.847 --> 00:02:36.661
Some kids like Hannah
Albright work on their own

00:02:36.661 --> 00:02:39.655
following online exercises
as they do calculations

00:02:39.655 --> 00:02:42.434
on a whiteboard or with pencil and paper.

00:02:42.434 --> 00:02:48.619
Hannah: Nine minus seven.
Then that would be 83.

00:02:48.619 --> 00:02:50.333
It's like an online teacher.

00:02:50.333 --> 00:02:51.909
They can help you anywhere.

00:02:51.909 --> 00:02:53.125
Larry: You can go as fast as you want.

00:02:53.125 --> 00:02:57.936
Hannah: Yeah. (students
talking) We don't have to wait.

00:02:57.936 --> 00:02:59.651
Next door in Kelly Rafferty's class

00:02:59.651 --> 00:03:01.435
kids put their name on the board.

00:03:01.435 --> 00:03:03.033
Some want help on a concept.

00:03:03.033 --> 00:03:04.434
Others offer to teach something

00:03:04.434 --> 00:03:05.858
that they've mastered.

00:03:05.858 --> 00:03:07.493
Rafferty says that Kahn helps her deal

00:03:07.493 --> 00:03:09.836
with the big class sizes that even

00:03:09.836 --> 00:03:12.134
this affluent district is facing.

00:03:12.134 --> 00:03:14.340
Kelly: Kahn has definitely helped me reach

00:03:14.340 --> 00:03:17.752
all of the students so when
you have that many kids,

00:03:17.752 --> 00:03:20.728
you end up sort of teaching to the middle.

00:03:20.728 --> 00:03:24.840
Kahn has allowed me to reach
the lowest and the highest.

00:03:24.840 --> 00:03:27.730
Larry: Much more than a
textbook would Rafferty says.

00:03:27.730 --> 00:03:29.663
Kahn Academy also offers a kind

00:03:29.663 --> 00:03:31.230
of back office function.

00:03:31.230 --> 00:03:32.750
While kids are working independently

00:03:32.750 --> 00:03:35.334
on exercises, Rafferty checks her computer

00:03:35.334 --> 00:03:36.934
to see how they're doing.

00:03:36.934 --> 00:03:38.244
Kelly: With this I can
look at all the kids,

00:03:38.244 --> 00:03:41.250
and I can see who's struggling, who's not,

00:03:41.250 --> 00:03:44.082
which they've passed, how
many it took for them to pass.

00:03:44.082 --> 00:03:46.334
It gives me a lot more information.

00:03:46.334 --> 00:03:47.534
Larry: Information she can use

00:03:47.534 --> 00:03:49.837
to see who needs extra help.

00:03:49.837 --> 00:03:52.829
Santa Rita Elementary
Principal Sandra McGonagle says

00:03:52.829 --> 00:03:54.440
for her the Khan Academy has some

00:03:54.440 --> 00:03:57.250
big advantages over other math materials.

00:03:57.250 --> 00:03:59.633
Sandra: One, it's free, and it's created

00:03:59.633 --> 00:04:02.833
by people who really just love math.

00:04:02.833 --> 00:04:04.634
The quality is so great.

00:04:04.634 --> 00:04:06.164
Larry: After this year's pilot program,

00:04:06.164 --> 00:04:07.544
the Los Altos schools are going

00:04:07.544 --> 00:04:10.182
to start using Khan Academy district wide

00:04:10.182 --> 00:04:12.411
for 5th and 6th grade.

00:04:12.411 --> 00:04:13.830
The Khan Academy is hoping to see how

00:04:13.830 --> 00:04:15.305
this experiment plays out

00:04:15.305 --> 00:04:17.138
in a wider range of schools.

00:04:17.138 --> 00:04:20.315
They expand to at least
10 new schools next year.

00:04:20.315 --> 00:04:22.641
Math class may never be the same.

00:04:22.641 --> 00:04:26.567
Student: I'm okay. (students talking)

00:04:26.567 --> 00:04:29.000
Larry: Larry Abramson, NPR News.


Radical equivalent to rational exponents 2

WEBVTT
Kind: captions
Language: en

00:00:00.906 --> 00:00:02.396
Rewrite the radical expression

00:00:02.396 --> 00:00:05.921
using rational exponents and simplify.

00:00:05.921 --> 00:00:09.915
So here we have the fourth root of 5 a to the fourth

00:00:09.915 --> 00:00:11.587
b to the twelfth power.

00:00:11.587 --> 00:00:16.091
The key thing to realize here is the fourth root of something

00:00:16.091 --> 00:00:19.853
is same thing as something to the one fourth power.

00:00:19.853 --> 00:00:23.475
Or in particular, or in general, the nth root something

00:00:23.475 --> 00:00:27.562
is the same thing as that something to the 1 over n power.

00:00:27.562 --> 00:00:29.745
So we can just apply that over here:

00:00:29.745 --> 00:00:30.665
the fourth root of all of this

00:00:30.665 --> 00:00:35.133
is equal to 5 a to the 4th, b to the 12th power,

00:00:35.133 --> 00:00:38.166
all of that to the one fourth power.

00:00:38.166 --> 00:00:41.215
And then we also know if we take the product of things,

00:00:41.215 --> 00:00:43.491
and then raise them to some exponent,

00:00:43.491 --> 00:00:46.928
that's the same thing as raising each of the terms in the product

00:00:46.928 --> 00:00:47.998
to the exponent first,

00:00:47.998 --> 00:00:50.550
or each of the things that we're taking the product of

00:00:50.550 --> 00:00:52.825
to that exponent, and then multiplying.

00:00:52.825 --> 00:00:55.057
So let's do that.

00:00:55.057 --> 00:00:57.302
So 5, so this is the same thing as

00:00:57.302 --> 00:00:59.039
5 to the 1/4 power

00:00:59.039 --> 00:01:02.430
times a to the fourth to the 1/4 power

00:01:02.430 --> 00:01:07.222
times b to the 12th to the 1/4 power.

00:01:07.222 --> 00:01:10.147
Now 5 to the 1/4,

00:01:10.147 --> 00:01:11.329
I don't know what that is,

00:01:11.329 --> 00:01:13.605
so I'll just keep that as the cube root,

00:01:13.605 --> 00:01:15.805
well, we could leave it as 5 to the 1/4,

00:01:15.805 --> 00:01:17.876
and that's not not simplified.

00:01:17.876 --> 00:01:23.027
or we can just rewrite it again as the fourth root of 5.

00:01:23.058 --> 00:01:25.446
a to the fourth to the 1/4 power:

00:01:25.446 --> 00:01:28.740
if your raise something to a power and then another power,

00:01:28.740 --> 00:01:30.151
and raise that to another power,

00:01:30.151 --> 00:01:35.267
that's equivalent of raising a to the four times 1/4 power.

00:01:35.267 --> 00:01:36.329
So let me just write that down.

00:01:36.329 --> 00:01:41.200
This is, so times a to four times 1/4 power.

00:01:41.200 --> 00:01:44.234
And then finally, this right over here,

00:01:44.234 --> 00:01:46.928
using the same exact exponent property,

00:01:46.928 --> 00:01:50.504
this is b to the 12th times 1/4 power.

00:01:50.504 --> 00:01:53.151
So all of this simplifies to,

00:01:53.151 --> 00:01:55.241
and I'll change the order here,

00:01:55.241 --> 00:01:59.112
so you have the fourth root of 5,

00:01:59.112 --> 00:02:02.903
and then you have a to the fourth times 1/4 power

00:02:02.903 --> 00:02:05.368
so that's just, this simplifies to a to the first power

00:02:05.368 --> 00:02:07.059
which is really just the same thing as a.

00:02:07.075 --> 00:02:08.902
So that's just a.

00:02:08.902 --> 00:02:13.882
And then we have b to the 12 times 1/4 power.

00:02:13.882 --> 00:02:16.454
Well, 12 times 1/4 is just three.

00:02:16.454 --> 00:02:19.036
So that's b to the third power.

00:02:19.036 --> 99:59:59.999
So it's a, b to the third power times the fourth root of 5.


More rational exponents and exponent laws

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.470
Simplify the expression
using rational exponents.

00:00:04.470 --> 00:00:06.865
So we have the cubed root of
all of this stuff over here,

00:00:06.865 --> 00:00:08.490
which would be the
same thing, or which

00:00:08.490 --> 00:00:11.140
is the same thing by definition,
as all of this stuff--

00:00:11.140 --> 00:00:15.180
64a to the sixth, b to
the third, c to the ninth,

00:00:15.180 --> 00:00:18.440
raised to the 1/3 power.

00:00:18.440 --> 00:00:21.082
These are equivalent
by definition.

00:00:21.082 --> 00:00:23.040
And if we take the product
of a bunch of things

00:00:23.040 --> 00:00:24.581
and raise them to
an exponent, that's

00:00:24.581 --> 00:00:26.690
the same thing as raising
each of these things

00:00:26.690 --> 00:00:28.940
that we're taking the product
of to the exponent first

00:00:28.940 --> 00:00:29.980
and then multiplying.

00:00:29.980 --> 00:00:36.020
So this is going to be equal
to 64 to the 1/3 power times

00:00:36.020 --> 00:00:40.340
a to the sixth to
the 1/3 power times

00:00:40.340 --> 00:00:43.990
b to the third to
the 1/3 power times

00:00:43.990 --> 00:00:47.210
c to the ninth to the 1/3 power.

00:00:47.210 --> 00:00:49.497
And we've seen this in
several examples already.

00:00:49.497 --> 00:00:51.080
Well first let's try
to simplify this.

00:00:51.080 --> 00:00:52.650
64 to the 1/3.

00:00:52.650 --> 00:00:54.170
You might recognize
that already,

00:00:54.170 --> 00:00:57.146
but we can factor 64
as-- well, I wouldn't

00:00:57.146 --> 00:00:59.020
do a pure prime
factorization, because you'll

00:00:59.020 --> 00:01:00.545
see it's 4 to the third power.

00:01:00.545 --> 00:01:02.420
Actually, you could
verify that for yourself.

00:01:02.420 --> 00:01:03.920
Or I encourage
you, if that's not

00:01:03.920 --> 00:01:07.410
obvious to see that this is
4 to the third power, 64,

00:01:07.410 --> 00:01:08.980
multiply it out for yourself.

00:01:08.980 --> 00:01:10.710
Or if you're like,
how did you know that?

00:01:10.710 --> 00:01:12.190
Do the prime factorization.

00:01:12.190 --> 00:01:13.260
And what you're going
to do is if you--

00:01:13.260 --> 00:01:14.110
well, let me just show you.

00:01:14.110 --> 00:01:15.526
If you do the prime
factorization,

00:01:15.526 --> 00:01:17.360
you're going to have 2 times 32.

00:01:17.360 --> 00:01:19.526
32 is 2 times 16.

00:01:19.526 --> 00:01:21.570
16 is 2 times 8.

00:01:21.570 --> 00:01:23.650
8 is 2 times 4.

00:01:23.650 --> 00:01:25.690
And then 4 is 2 times 2.

00:01:25.690 --> 00:01:28.920
So you can have 2 multiplied
by itself three times,

00:01:28.920 --> 00:01:32.050
or you could have it-- a 2
multiplied by itself three

00:01:32.050 --> 00:01:33.280
times and that happens twice.

00:01:33.280 --> 00:01:38.750
Or you could have
4 times 4 times 4.

00:01:38.750 --> 00:01:41.590
So if it immediately
didn't jump into your head

00:01:41.590 --> 00:01:43.840
that 64 is 4 to the
third power, you

00:01:43.840 --> 00:01:47.250
can literally do a brute
force prime factorization here

00:01:47.250 --> 00:01:50.160
and see that you have
6-- you could factor this

00:01:50.160 --> 00:01:54.170
into 2 to the sixth power,
or the same thing as 4

00:01:54.170 --> 00:01:55.000
to the third power.

00:01:55.000 --> 00:01:57.960
So I'll let you think
about that a little bit.

00:01:57.960 --> 00:02:01.800
But this right here simplifies
that 64 is 4 to the third power

00:02:01.800 --> 00:02:03.480
and you raise that
to the 1/3 power.

00:02:03.480 --> 00:02:04.630
That is the same thing.

00:02:04.630 --> 00:02:05.740
Actually, let me
write it that way.

00:02:05.740 --> 00:02:07.240
That could make it
more interesting.

00:02:07.240 --> 00:02:10.860
So this right here, this
is 4 the third power.

00:02:10.860 --> 00:02:12.820
So I have 4 to the
third power and then

00:02:12.820 --> 00:02:15.290
that raised to the 1/3 power.

00:02:15.290 --> 00:02:17.380
So if I want to rewrite
that term right over here,

00:02:17.380 --> 00:02:20.550
that's going to be
four to the 3/3 power.

00:02:20.550 --> 00:02:22.610
I'm literally
multiplying 3 times 1/3.

00:02:22.610 --> 00:02:25.760
4 to the third and
then that to the 1/3.

00:02:25.760 --> 00:02:29.370
Then over here, I have a to
the sixth and then that the 1/3

00:02:29.370 --> 00:02:30.190
power.

00:02:30.190 --> 00:02:33.620
That is going to be a to
the 6 times 1/3 power,

00:02:33.620 --> 00:02:36.130
or a to the 6/3 power.

00:02:36.130 --> 00:02:39.710
Then I have b to the third and
then that raised to the 1/3,

00:02:39.710 --> 00:02:41.600
so that's going to
be-- we literally

00:02:41.600 --> 00:02:43.230
can multiply exponents here.

00:02:43.230 --> 00:02:46.440
So we have b to
the 3 over 3 power.

00:02:46.440 --> 00:02:49.660
And then finally, we
have c to the ninth

00:02:49.660 --> 00:02:51.520
and then that raised
to the 1/3 power.

00:02:51.520 --> 00:02:56.100
So that is c to the 9/3 power.

00:02:56.100 --> 00:03:01.010
And if you simplify it, this
first term here, 4 to the 3/3,

00:03:01.010 --> 00:03:02.400
that's just 4 to
the first power.

00:03:02.400 --> 00:03:04.310
So that's just 4.

00:03:04.310 --> 00:03:05.710
We have a to the 6/3.

00:03:05.710 --> 00:03:09.500
6/3 is just 2, so that
just becomes a squared.

00:03:09.500 --> 00:03:14.200
B to the 3/3 power, that's
just b to the first or b.

00:03:14.200 --> 00:03:16.900
And then finally, c
to the 9/3, that's

00:03:16.900 --> 00:03:18.960
just c to the third power.

00:03:18.960 --> 00:03:21.280
So we write c to the third here.

00:03:21.280 --> 00:03:23.997
And we are done.


Simplifying cube root expressions (two variables)

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:05.060
Simplify the cube root
of 125 x to the sixth y

00:00:05.060 --> 00:00:06.280
to the third power.

00:00:06.280 --> 00:00:07.890
So taking the cube
root of something

00:00:07.890 --> 00:00:10.760
is the same thing as raising
that something to the 1/3

00:00:10.760 --> 00:00:11.330
power.

00:00:11.330 --> 00:00:16.720
So this is equal to
125 x to the sixth y

00:00:16.720 --> 00:00:21.430
to the third power
raised to the 1/3 power.

00:00:21.430 --> 00:00:23.480
And if we take a product
of a bunch of stuff

00:00:23.480 --> 00:00:25.320
and raise that to
the 1/3 power, that's

00:00:25.320 --> 00:00:28.140
the same thing as individually
raising each of the things

00:00:28.140 --> 00:00:30.750
to the 1/3 power and
then taking the product.

00:00:30.750 --> 00:00:34.310
So this is going
to be equal to 125

00:00:34.310 --> 00:00:41.970
to the 1/3 power times x to the
sixth to the 1/3 power times

00:00:41.970 --> 00:00:46.390
y to the third to the 1/3 power.

00:00:46.390 --> 00:00:49.990
And then we can think about how
we can simplify each of these.

00:00:49.990 --> 00:00:52.109
What's 125 to the 1/3?

00:00:52.109 --> 00:00:53.650
Well, let's just
factor and see if we

00:00:53.650 --> 00:00:56.100
can have at least three
prime factors of something

00:00:56.100 --> 00:00:58.860
and maybe more than
one prime factor

00:00:58.860 --> 00:01:00.460
that shows up three times.

00:01:00.460 --> 00:01:03.690
So 125 is 5 times 25.

00:01:03.690 --> 00:01:06.780
25 is 5 times 5.

00:01:06.780 --> 00:01:10.310
So 125 really is
5 times 5 times 5.

00:01:10.310 --> 00:01:14.581
So if you multiply 5 times
itself three times you get 125.

00:01:14.581 --> 00:01:18.410
125 to the 1/3 power
is going to be 5.

00:01:18.410 --> 00:01:22.470
So this is going to
simplify to 5 times.

00:01:22.470 --> 00:01:24.820
And then x to the sixth
to the 1/3 power--

00:01:24.820 --> 00:01:26.944
we saw this in a
previous example--

00:01:26.944 --> 00:01:28.360
if you raise a
base to an exponent

00:01:28.360 --> 00:01:30.740
and then raise that whole
thing to another exponent,

00:01:30.740 --> 00:01:33.400
you can take the product
of the two exponents.

00:01:33.400 --> 00:01:38.480
So 6 times 1/3 is 6/3 or 2.

00:01:38.480 --> 00:01:40.660
So this part right
over here simplifies

00:01:40.660 --> 00:01:47.580
to x to the sixth divided
by 3 power or x squared.

00:01:47.580 --> 00:01:50.440
And then finally over
here, same principle.

00:01:50.440 --> 00:01:54.280
Raising y to the third power,
and then that to the 1/3 power.

00:01:54.280 --> 00:01:56.270
So that's going to
be y to the 3 times

00:01:56.270 --> 00:02:00.150
1/3 power, or y to
the first power.

00:02:00.150 --> 00:02:02.360
And then times y.

00:02:02.360 --> 00:02:03.310
And we are done.

00:02:03.310 --> 00:02:05.685
And if you don't want to write
this little multiplication

00:02:05.685 --> 00:02:12.890
here, you could just write
this as 5x squared y.

00:02:12.890 --> 00:02:15.197
And we have simplified.


Radical equivalent to rational exponents

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.420
Express d times 4d to the
fourth to the 1/2 power

00:00:05.420 --> 00:00:07.120
as a radical expression.

00:00:07.120 --> 00:00:09.450
So the key thing
to realize here is

00:00:09.450 --> 00:00:14.410
that if I take anything to
the 1/2 power, by definition,

00:00:14.410 --> 00:00:17.520
this is the same thing as
taking the principal root

00:00:17.520 --> 00:00:18.920
of that anything.

00:00:18.920 --> 00:00:20.170
So let's just apply that here.

00:00:20.170 --> 00:00:23.570
We have 4d to the fourth
raised to the 1/2 power.

00:00:23.570 --> 00:00:25.000
So this part right
over here-- let

00:00:25.000 --> 00:00:26.510
me do it in two
different colors.

00:00:26.510 --> 00:00:28.460
So I have d right over there.

00:00:28.460 --> 00:00:31.910
And then I have 4d to the
fourth raised to the 1/2 power,

00:00:31.910 --> 00:00:37.777
which, by definition, is
the principal root of 4d

00:00:37.777 --> 00:00:38.360
to the fourth.

00:00:38.360 --> 00:00:40.776
And we explore in other videos
why this all works out well

00:00:40.776 --> 00:00:43.100
with exponent properties.

00:00:43.100 --> 00:00:44.950
And we know how to
simplify this right here.

00:00:44.950 --> 00:00:46.500
We're looking for
the principal root

00:00:46.500 --> 00:00:48.250
of-- let me write it this way.

00:00:48.250 --> 00:00:53.740
This is the same thing as
d times the principal root

00:00:53.740 --> 00:00:58.075
of 4 times the principal
root of d to the fourth.

00:01:01.820 --> 00:01:06.380
The principal root of 4 is 2.

00:01:06.380 --> 00:01:08.040
And we have that d out front.

00:01:08.040 --> 00:01:10.900
And the principal root
of d to the fourth power

00:01:10.900 --> 00:01:14.400
can be viewed as the positive
square root of d to the fourth,

00:01:14.400 --> 00:01:16.392
and that is d squared.

00:01:16.392 --> 00:01:18.350
And we don't have to put
an absolute value here

00:01:18.350 --> 00:01:21.830
because even if d
is a negative value,

00:01:21.830 --> 00:01:25.540
d squared will still
be a positive value.

00:01:25.540 --> 00:01:27.230
If this was a d
squared here, then we

00:01:27.230 --> 00:01:30.410
would have to think about the
absolute value right over here.

00:01:30.410 --> 00:01:33.400
And then just to simplify this,
we have a d times a 2 times

00:01:33.400 --> 00:01:34.670
a d squared.

00:01:34.670 --> 00:01:37.230
That's-- multiply the d's
times each other, same base--

00:01:37.230 --> 00:01:38.820
that becomes d to the third.

00:01:38.820 --> 00:01:44.212
So this becomes 2 times
d to the third power.

00:01:44.212 --> 00:01:45.920
Now, another way you
could have done this

00:01:45.920 --> 00:01:48.790
is to just straight use
the exponent properties.

00:01:48.790 --> 00:01:51.880
You could have said that that
is the same thing as d times--

00:01:51.880 --> 00:01:56.260
and inside the parentheses, if
you take the product of things

00:01:56.260 --> 00:01:58.190
and then raise them
to an exponent, that's

00:01:58.190 --> 00:02:00.010
the same thing as
raising each of them

00:02:00.010 --> 00:02:03.010
individually to the exponent
and then taking the product.

00:02:03.010 --> 00:02:04.410
So that's the same
thing-- so I'm

00:02:04.410 --> 00:02:06.560
going to rewrite this
part right over here

00:02:06.560 --> 00:02:14.170
as 4 to the 1/2 power times d
to the fourth to the 1/2 power.

00:02:14.170 --> 00:02:16.700
And 4 to the 1/2 power,
this is literally just

00:02:16.700 --> 00:02:20.110
the principal square
root of 4, which is 2.

00:02:20.110 --> 00:02:23.110
And d to the fourth
to the 1/2 power.

00:02:23.110 --> 00:02:25.190
This is, again, you can
use an exponent property.

00:02:25.190 --> 00:02:27.910
If I have something
raised to some power

00:02:27.910 --> 00:02:30.000
and then I have it
raised to another power,

00:02:30.000 --> 00:02:33.070
this is the same thing
as that same base raised

00:02:33.070 --> 00:02:35.200
to the product of these powers.

00:02:35.200 --> 00:02:37.120
And we prove it in other videos.

00:02:37.120 --> 00:02:39.730
So this right over here, this
expression right over here

00:02:39.730 --> 00:02:44.990
is going to be equal to d
to the 4 times 1/2 power.

00:02:44.990 --> 00:02:46.960
Well, what's 4 times 1/2?

00:02:46.960 --> 00:02:48.660
Well, that's just 2.

00:02:48.660 --> 00:02:52.880
So this simplifies, once again,
to d times 2 times d squared,

00:02:52.880 --> 00:02:56.140
which is 2d to the third power.


Simplifying cube roots

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:05.110
We're asked to simplify the
cube root of 27a squared times

00:00:05.110 --> 00:00:08.590
b to the fifth times
c to the third power.

00:00:08.590 --> 00:00:12.090
And the goal, whenever
you try to just simplify

00:00:12.090 --> 00:00:13.940
a cube root like
this, is we want

00:00:13.940 --> 00:00:16.430
to look at the parts of
this expression over here

00:00:16.430 --> 00:00:18.660
that are perfect cubes,
that are something

00:00:18.660 --> 00:00:20.170
raised to the third power.

00:00:20.170 --> 00:00:22.190
Then we can take just
the cube root of those,

00:00:22.190 --> 00:00:24.579
essentially taking them
out of the radical sign,

00:00:24.579 --> 00:00:26.120
and then leaving
everything else that

00:00:26.120 --> 00:00:27.930
is not a perfect
cube inside of it.

00:00:27.930 --> 00:00:29.390
So let's see what we can do.

00:00:29.390 --> 00:00:31.410
So first of all, 27--
you may or may not

00:00:31.410 --> 00:00:33.350
already recognize this
as a perfect cube.

00:00:33.350 --> 00:00:34.800
If you don't already
recognize it,

00:00:34.800 --> 00:00:36.508
you can actually do
a prime factorization

00:00:36.508 --> 00:00:38.040
and see it's a perfect cube.

00:00:38.040 --> 00:00:44.070
27 is 3 times 9,
and 9 is 3 times 3.

00:00:44.070 --> 00:00:47.990
So 27-- its prime factorization
is 3 times 3 times 3.

00:00:47.990 --> 00:00:51.780
So it's the exact same thing
as 3 to the third power.

00:00:51.780 --> 00:00:53.860
So let's rewrite this
whole expression down here.

00:00:53.860 --> 00:00:56.140
But let's write it in terms of
things that are perfect cubes

00:00:56.140 --> 00:00:57.098
and things that aren't.

00:00:57.098 --> 00:01:01.900
So 27 can be just rewritten
as 3 to the third power.

00:01:01.900 --> 00:01:05.710
Then you have a squared--
clearly not a perfect cube.

00:01:05.710 --> 00:01:07.150
a to the third would have been.

00:01:07.150 --> 00:01:08.617
So we're just going
to write this--

00:01:08.617 --> 00:01:09.700
let me write it over here.

00:01:09.700 --> 00:01:11.050
We can switch the order
here because we just

00:01:11.050 --> 00:01:13.750
have a bunch of things being
multiplied by each other.

00:01:13.750 --> 00:01:16.470
So I'll write the a
squared over here.

00:01:16.470 --> 00:01:21.590
b to the fifth is not a
perfect cube by itself,

00:01:21.590 --> 00:01:23.580
but it can be expressed
as the product

00:01:23.580 --> 00:01:26.600
of a perfect cube and
another thing. b to the fifth

00:01:26.600 --> 00:01:30.240
is the exact same thing as
b to the third power times

00:01:30.240 --> 00:01:31.427
b to the second power.

00:01:31.427 --> 00:01:33.510
If you want to see that
explicitly, b to the fifth

00:01:33.510 --> 00:01:38.800
is b times b times
b times b times b.

00:01:38.800 --> 00:01:42.810
So the first three are
clearly b to the third power.

00:01:42.810 --> 00:01:45.400
And then you have b to
the second power after it.

00:01:45.400 --> 00:01:47.850
So we can rewrite b to
the fifth as the product

00:01:47.850 --> 00:01:49.680
of a perfect cube.

00:01:49.680 --> 00:01:51.542
So I'll write b
to the third-- let

00:01:51.542 --> 00:01:54.340
me do that in that
same purple color.

00:01:54.340 --> 00:01:57.540
So we have b to the
third power over here.

00:01:57.540 --> 00:01:59.620
And then it's b to the
third times b squared.

00:01:59.620 --> 00:02:01.310
So I'll write the b
squared over here.

00:02:01.310 --> 00:02:03.851
And we're assuming we're going
to multiply all of this stuff.

00:02:03.851 --> 00:02:08.029
And then finally, we
have-- I'll do in blue-- c

00:02:08.029 --> 00:02:09.190
to the third power.

00:02:09.190 --> 00:02:11.205
Clearly, this is a perfect cube.

00:02:11.205 --> 00:02:12.850
It is c cubed.

00:02:12.850 --> 00:02:15.030
It is c to the third power.

00:02:15.030 --> 00:02:16.090
So I'll put it over here.

00:02:16.090 --> 00:02:17.960
So this is c to the third power.

00:02:17.960 --> 00:02:21.680
And of course, we still have
that overarching radical sign.

00:02:21.680 --> 00:02:25.770
So we're still trying to take
the cube root of all of this.

00:02:25.770 --> 00:02:27.640
And we know from our
exponent properties,

00:02:27.640 --> 00:02:29.630
or we could say from
our radical properties,

00:02:29.630 --> 00:02:31.071
that this is the
exact same thing.

00:02:31.071 --> 00:02:33.070
That taking the cube root
of all of these things

00:02:33.070 --> 00:02:34.880
is the same as
taking the cube root

00:02:34.880 --> 00:02:38.230
of these individual factors
and then multiplying them.

00:02:38.230 --> 00:02:40.739
So this is the same
thing as the cube root--

00:02:40.739 --> 00:02:42.530
and I could separate
them out individually.

00:02:42.530 --> 00:02:45.970
Or I could say the cube
root of 3 to the third b

00:02:45.970 --> 00:02:47.280
to the third c to the third.

00:02:47.280 --> 00:02:48.636
Actually, let's do it both ways.

00:02:48.636 --> 00:02:50.010
So I'll take them
out separately.

00:02:50.010 --> 00:02:53.000
So this is the same thing
as the cube root of 3

00:02:53.000 --> 00:02:57.730
to the third times the cube
root-- I'll write them all in.

00:02:57.730 --> 00:03:01.120
Let me color-code it so we don't
get confused-- times the cube

00:03:01.120 --> 00:03:08.840
root of b to the third
times the cube root of c

00:03:08.840 --> 00:03:13.430
to the third times
the cube root--

00:03:13.430 --> 00:03:14.900
and I'll just group
these two guys

00:03:14.900 --> 00:03:17.680
together just because we're not
going to be able to simplify it

00:03:17.680 --> 00:03:22.919
any more-- times the cube
root of a squared b squared.

00:03:22.919 --> 00:03:24.710
I'll keep the colors
consistent while we're

00:03:24.710 --> 00:03:27.740
trying to figure
out what's what.

00:03:27.740 --> 00:03:29.950
And I could have said that
this is times the cube

00:03:29.950 --> 00:03:32.150
root of a squared times
the cube root of b squared,

00:03:32.150 --> 00:03:33.524
but that won't
simplify anything,

00:03:33.524 --> 00:03:35.350
so I'll just leave
these like this.

00:03:35.350 --> 00:03:37.070
And so we can look at
these individually.

00:03:37.070 --> 00:03:39.970
The cube root of 3 to the
third, or the cube root of 27--

00:03:39.970 --> 00:03:42.000
well, that's clearly
just going to be-- I

00:03:42.000 --> 00:03:44.360
want to do that in
that yellow color--

00:03:44.360 --> 00:03:46.686
this is clearly
just going to be 3.

00:03:46.686 --> 00:03:49.260
3 to the third power is
3 to the third power,

00:03:49.260 --> 00:03:50.850
or it's equal to 27.

00:03:50.850 --> 00:03:53.180
This term right over
here, the cube root

00:03:53.180 --> 00:03:56.820
of b to the third--
well, that's just b.

00:03:56.820 --> 00:04:00.160
And the cube root
of c to the third,

00:04:00.160 --> 00:04:02.050
well, that is
clearly-- I want to do

00:04:02.050 --> 00:04:05.990
that in that-- that
is clearly just c.

00:04:05.990 --> 00:04:12.560
So our whole expression has
simplified to 3 times b times

00:04:12.560 --> 00:04:20.430
c times the cube root
of a squared b squared.

00:04:26.750 --> 00:04:27.837
And we're done.

00:04:27.837 --> 00:04:29.420
And I just want to
do one other thing,

00:04:29.420 --> 00:04:31.370
just because I did mention
that I would do it.

00:04:31.370 --> 00:04:32.790
We could simplify it this way.

00:04:32.790 --> 00:04:37.420
Or we could recognize that
this expression right over here

00:04:37.420 --> 00:04:43.714
can be written as 3bc
to the third power.

00:04:43.714 --> 00:04:45.630
And if I take three
things to the third power,

00:04:45.630 --> 00:04:46.930
and I'm multiplying it,
that's the same thing

00:04:46.930 --> 00:04:48.699
as multiplying them
first and then raising

00:04:48.699 --> 00:04:49.490
to the third power.

00:04:49.490 --> 00:04:51.620
It comes straight out of
our exponent properties.

00:04:51.620 --> 00:04:53.960
And so we can rewrite
this as the cube

00:04:53.960 --> 00:05:01.580
root of all of this times
the cube root of a squared b

00:05:01.580 --> 00:05:02.701
squared.

00:05:02.701 --> 00:05:04.200
And so the cube
root of all of this,

00:05:04.200 --> 00:05:07.890
of 3bc to the third power, well,
that's just going to be 3bc,

00:05:07.890 --> 00:05:11.742
and then multiplied by the cube
root of a squared b squared.

00:05:11.742 --> 00:05:13.950
I didn't take the trouble
to color-code it this time,

00:05:13.950 --> 00:05:15.910
because we already figured
out one way to solve it.

00:05:15.910 --> 00:05:17.451
But hopefully, that
also makes sense.

00:05:17.451 --> 00:05:19.940
We could have done
this either way.

00:05:19.940 --> 00:05:23.603
But the important thing is
that we get that same answer.


Cube root of a negative number (example)

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:05.580
We are asked to find the
cube root of negative 512.

00:00:05.580 --> 00:00:08.460
Or another way to think about
it is if I have some number,

00:00:08.460 --> 00:00:13.740
and it is equal to the
cube root of negative 512,

00:00:13.740 --> 00:00:16.040
this just means that
if I take that number

00:00:16.040 --> 00:00:20.900
and I raise it to the third
power, then I get negative 512.

00:00:20.900 --> 00:00:24.960
And if it doesn't jump out
at you immediately what

00:00:24.960 --> 00:00:27.120
this is the cube
of, or what we have

00:00:27.120 --> 00:00:30.320
to raise to the third
power to get negative 512,

00:00:30.320 --> 00:00:33.530
the best thing to do is to just
do a prime factorization of it.

00:00:33.530 --> 00:00:35.460
And before we do a prime
factorization of it

00:00:35.460 --> 00:00:38.840
to see which of these factors
show up at least three times,

00:00:38.840 --> 00:00:41.450
let's at least think about the
negative part a little bit.

00:00:41.450 --> 00:00:44.350
So negative 512, that's
the same thing-- so

00:00:44.350 --> 00:00:45.850
let me rewrite the
expression-- this

00:00:45.850 --> 00:00:52.170
is the same thing as the cube
root of negative 1 times 512,

00:00:52.170 --> 00:00:59.540
which is the same thing as
the cube root of negative 1

00:00:59.540 --> 00:01:04.771
times the cube root of 512.

00:01:04.771 --> 00:01:06.770
And this one's pretty
straightforward to answer.

00:01:06.770 --> 00:01:08.770
What number, when I raise
it to the third power,

00:01:08.770 --> 00:01:10.020
do I get negative 1?

00:01:10.020 --> 00:01:11.440
Well, I get negative 1.

00:01:11.440 --> 00:01:13.560
This right here is negative 1.

00:01:13.560 --> 00:01:16.650
Negative 1 to the third power
is equal to negative 1 times

00:01:16.650 --> 00:01:20.470
negative 1 times negative 1,
which is equal to negative 1.

00:01:20.470 --> 00:01:23.800
So the cube root of
negative 1 is negative 1.

00:01:23.800 --> 00:01:27.890
So it becomes negative 1
times this business right

00:01:27.890 --> 00:01:32.172
here, times the
cube root of 512.

00:01:32.172 --> 00:01:33.630
And let's think
what this might be.

00:01:33.630 --> 00:01:36.500
So let's do the
prime factorization.

00:01:36.500 --> 00:01:41.019
So 512 is 2 times 256.

00:01:41.019 --> 00:01:46.220
256 is 2 times 128.

00:01:46.220 --> 00:01:49.320
128 is 2 times 64.

00:01:49.320 --> 00:01:51.240
We already see a 2 three times.

00:01:51.240 --> 00:01:56.890
64 is 2 times 32.

00:01:56.890 --> 00:01:58.740
32 is 2 times 16.

00:01:58.740 --> 00:02:00.310
We're getting a
lot of twos here.

00:02:00.310 --> 00:02:05.930
16 is 2 times 8.

00:02:05.930 --> 00:02:09.539
8 is 2 times 4.

00:02:09.539 --> 00:02:10.960
And 4 is 2 times 2.

00:02:10.960 --> 00:02:13.010
So we got a lot of twos.

00:02:13.010 --> 00:02:14.966
So essentially, if
you multiply 2 one,

00:02:14.966 --> 00:02:19.290
two, three, four, five, six,
seven, eight, nine times,

00:02:19.290 --> 00:02:23.184
you're going to get 512, or
2 to the ninth power is 512.

00:02:23.184 --> 00:02:24.600
And that by itself
should give you

00:02:24.600 --> 00:02:26.025
a clue of what the cube root is.

00:02:26.025 --> 00:02:27.400
But another way
to think about it

00:02:27.400 --> 00:02:31.600
is, can we find-- there's
definitely three twos here.

00:02:31.600 --> 00:02:34.290
But can we find
three groups of twos,

00:02:34.290 --> 00:02:37.480
or we could also find--
let me look at it this way.

00:02:37.480 --> 00:02:42.160
We can find three groups
of two twos over here.

00:02:42.160 --> 00:02:43.780
So that's 2 times 2 is 4.

00:02:43.780 --> 00:02:44.620
2 times 2 is 4.

00:02:44.620 --> 00:02:47.490
So definitely 4 multiplied
by itself three times

00:02:47.490 --> 00:02:48.986
is divisible into this.

00:02:48.986 --> 00:02:50.360
But even better,
it looks like we

00:02:50.360 --> 00:02:54.530
can get three groups
of three twos.

00:02:54.530 --> 00:02:59.365
So one group, two
groups, and three groups.

00:03:01.960 --> 00:03:05.600
So each of these groups, 2
times 2 times 2, that's 8.

00:03:05.600 --> 00:03:06.360
That is 8.

00:03:06.360 --> 00:03:07.790
This is 2 times 2 times 2.

00:03:07.790 --> 00:03:08.690
That's 8.

00:03:08.690 --> 00:03:10.400
And this is also
2 times 2 times 2.

00:03:10.400 --> 00:03:11.410
So that's 8.

00:03:11.410 --> 00:03:18.710
So we could write 512 as being
equal to 8 times 8 times 8.

00:03:18.710 --> 00:03:20.600
And so we can rewrite
this expression right

00:03:20.600 --> 00:03:24.940
over here as the cube
root of 8 times 8 times 8.

00:03:24.940 --> 00:03:28.400
So this is equal to
negative 1, or I could just

00:03:28.400 --> 00:03:31.090
put a negative sign
here, negative 1 times

00:03:31.090 --> 00:03:36.652
the cube root of
8 times 8 times 8.

00:03:36.652 --> 00:03:37.860
So we're asking our question.

00:03:37.860 --> 00:03:40.650
What number can we multiply
by itself three times,

00:03:40.650 --> 00:03:42.950
or to the third
power, to get 512,

00:03:42.950 --> 00:03:45.270
which is the same thing
as 8 times 8 times 8?

00:03:45.270 --> 00:03:46.770
Well, clearly this is 8.

00:03:46.770 --> 00:03:49.000
So the answer, this
part right over here,

00:03:49.000 --> 00:03:50.680
is just going to simplify to 8.

00:03:50.680 --> 00:03:54.800
And so our answer to this,
the cube root of negative 512,

00:03:54.800 --> 00:03:58.580
is negative 8.

00:03:58.580 --> 00:03:59.727
And we are done.

00:03:59.727 --> 00:04:00.810
And you could verify this.

00:04:00.810 --> 00:04:04.012
Multiply negative 8
times itself three times.

00:04:04.012 --> 00:04:04.970
Well, let's just do it.

00:04:04.970 --> 00:04:09.190
Negative 8 times negative
8 times negative 8.

00:04:09.190 --> 00:04:12.310
Negative 8 times negative
8 is positive 64.

00:04:12.310 --> 00:04:16.690
You multiply that times negative
8, you get negative 512.


Simplifying square roots comment response

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.840
What I want to do in
this video is resimplify

00:00:02.840 --> 00:00:06.610
this expression, 3 times the
principal root of 500 times x

00:00:06.610 --> 00:00:08.760
to the third, and take
into consideration

00:00:08.760 --> 00:00:10.730
some of the comments that
we got out on YouTube

00:00:10.730 --> 00:00:13.900
that actually give some
interesting perspective

00:00:13.900 --> 00:00:15.230
on how you could simplify this.

00:00:15.230 --> 00:00:17.702
So just as a quick review of
what we did in the last video,

00:00:17.702 --> 00:00:19.160
we said that this
is the same thing

00:00:19.160 --> 00:00:22.254
as 3 times the
principal root of 500.

00:00:22.254 --> 00:00:24.420
And I'm going to do it a
little bit different than I

00:00:24.420 --> 00:00:27.334
did in the last video, just
to make it interesting.

00:00:27.334 --> 00:00:28.750
This is 3 times
the principal root

00:00:28.750 --> 00:00:34.310
of 500 times the principal
root of x to the third.

00:00:34.310 --> 00:00:36.960
And 500-- we can
rewrite it, because 500

00:00:36.960 --> 00:00:38.140
is not a perfect square.

00:00:38.140 --> 00:00:43.040
We can rewrite 500
as 100 times 5.

00:00:43.040 --> 00:00:47.300
Or even better, we could rewrite
that as 10 squared times 5.

00:00:47.300 --> 00:00:49.680
10 squared is the
same thing as 100.

00:00:49.680 --> 00:00:52.840
So we can rewrite this first
part over here as 3 times

00:00:52.840 --> 00:00:55.180
the principal root
of 10 squared times

00:00:55.180 --> 00:01:02.860
5 times the principal
root of x squared times x.

00:01:02.860 --> 00:01:04.920
That's the same thing
as x to the third.

00:01:04.920 --> 00:01:06.106
Now, the one thing I'm
going to do here-- actually,

00:01:06.106 --> 00:01:07.490
I won't talk about
it just yet, of how

00:01:07.490 --> 00:01:08.650
we're going to do it
differently than we did it

00:01:08.650 --> 00:01:09.880
in the last video.

00:01:09.880 --> 00:01:11.990
This radical right
here can be rewritten

00:01:11.990 --> 00:01:15.441
as-- so this is going to
be 3 times the square root,

00:01:15.441 --> 00:01:16.940
or the principal
root, I should say,

00:01:16.940 --> 00:01:20.090
of 10 squared times
the square root of 5.

00:01:20.090 --> 00:01:22.860
If we take the square root
of the product of two things,

00:01:22.860 --> 00:01:25.401
it's the same thing as taking
the square root of each of them

00:01:25.401 --> 00:01:26.662
and then taking the product.

00:01:26.662 --> 00:01:28.120
And so then this
over here is going

00:01:28.120 --> 00:01:31.760
to be times the square root
of, or the principal root of,

00:01:31.760 --> 00:01:36.110
x squared times the
principal root of x.

00:01:36.110 --> 00:01:40.150
And the principal root
of 10 squared is 10.

00:01:40.150 --> 00:01:41.930
And then what I said
in the last video

00:01:41.930 --> 00:01:44.290
is that the principal
root of x squared

00:01:44.290 --> 00:01:48.990
is going to be the absolute
value of x, just in case x

00:01:48.990 --> 00:01:51.090
itself is a negative number.

00:01:51.090 --> 00:01:52.800
And so then if you
simplify all of this,

00:01:52.800 --> 00:01:55.685
you get 3 times
10, which is 30--

00:01:55.685 --> 00:01:57.560
and I'm just going to
switch the order here--

00:01:57.560 --> 00:02:00.470
times the absolute value of x.

00:02:00.470 --> 00:02:02.250
And then you have
the square root of 5,

00:02:02.250 --> 00:02:05.800
or the principal root of 5,
times the principal root of x.

00:02:05.800 --> 00:02:08.880
And this is just going to be
equal to the principal root

00:02:08.880 --> 00:02:10.160
of 5x.

00:02:10.160 --> 00:02:11.700
Taking the square
root of something

00:02:11.700 --> 00:02:14.199
and multiplying that times the
square root of something else

00:02:14.199 --> 00:02:16.910
is the same thing as just
taking the square root of 5x.

00:02:16.910 --> 00:02:19.600
So all of this simplified
down to 30 times

00:02:19.600 --> 00:02:22.740
the absolute value of x times
the principal root of 5x.

00:02:22.740 --> 00:02:24.550
And this is what we
got in the last video.

00:02:24.550 --> 00:02:26.470
And the interesting
thing here is,

00:02:26.470 --> 00:02:29.380
if we assume we're only
dealing with real numbers,

00:02:29.380 --> 00:02:32.770
the domain of x right
over here, the x's that

00:02:32.770 --> 00:02:34.950
will make this
expression defined

00:02:34.950 --> 00:02:39.720
in the real numbers--
then x has to be

00:02:39.720 --> 00:02:43.262
greater than or equal to 0.

00:02:43.262 --> 00:02:44.720
So maybe I could
write it this way.

00:02:44.720 --> 00:02:56.270
The domain here is that
x is any real number

00:02:56.270 --> 00:02:58.155
greater than or equal to 0.

00:02:58.155 --> 00:02:59.530
And the reason
why I say that is,

00:02:59.530 --> 00:03:03.737
if you put a negative number
in here and you cube it,

00:03:03.737 --> 00:03:05.570
you're going to get
another negative number.

00:03:05.570 --> 00:03:11.870
And then at least
in the real numbers,

00:03:11.870 --> 00:03:13.470
you won't get an actual value.

00:03:13.470 --> 00:03:16.120
You'll get a square root
of a negative number here.

00:03:16.120 --> 00:03:18.500
So if you make this-- if
you assume this right here,

00:03:18.500 --> 00:03:20.880
we're dealing with
the real numbers.

00:03:20.880 --> 00:03:22.680
We're not dealing with
any complex numbers.

00:03:22.680 --> 00:03:24.750
When you open it up
to complex numbers,

00:03:24.750 --> 00:03:28.464
then you can expand the
domain more broadly.

00:03:28.464 --> 00:03:30.130
But if you're dealing
with real numbers,

00:03:30.130 --> 00:03:33.410
you can say that x is going to
be greater than or equal to 0.

00:03:33.410 --> 00:03:35.240
And then the absolute
value of x is just

00:03:35.240 --> 00:03:38.110
going to be x, because it's not
going to be a negative number.

00:03:38.110 --> 00:03:41.490
And so if we're assuming
that the domain of x is--

00:03:41.490 --> 00:03:44.770
or if this expression is
going to be evaluatable,

00:03:44.770 --> 00:03:46.970
or it's going to have
a positive number,

00:03:46.970 --> 00:03:52.960
then this can be written as 30x
times the square root of 5x.

00:03:52.960 --> 00:03:55.350
If you had the situation
where we were dealing

00:03:55.350 --> 00:03:59.130
with complex
numbers-- and if you

00:03:59.130 --> 00:04:01.630
don't know what a complex number
is, or an imaginary number,

00:04:01.630 --> 00:04:03.310
don't worry too much about it.

00:04:03.310 --> 00:04:05.230
But if you were
dealing with those,

00:04:05.230 --> 00:04:07.740
then you would have to keep
the absolute value of x there.

00:04:07.740 --> 00:04:11.030
Because then this
would be defined

00:04:11.030 --> 00:04:15.260
for numbers that
are less than 0.


Simplifying square roots

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:05.800
Simplify 3 times the
principal root of 500x

00:00:05.800 --> 00:00:07.325
to the third power.

00:00:07.325 --> 00:00:08.700
So what we're
going to do here is

00:00:08.700 --> 00:00:11.910
try to break up this stuff that
we have under the radical sign

00:00:11.910 --> 00:00:14.145
into some things that
are perfect squares.

00:00:14.145 --> 00:00:16.770
And then we can separately take
the square root of those things

00:00:16.770 --> 00:00:19.025
and leave the left overs
inside of the radical sign.

00:00:19.025 --> 00:00:20.650
And the other thing
we have to remember

00:00:20.650 --> 00:00:23.440
is, principal square
root, this radical sign

00:00:23.440 --> 00:00:28.215
here really means the positive
the square root of a number.

00:00:28.215 --> 00:00:29.840
So let's think about
this a little bit.

00:00:29.840 --> 00:00:31.214
So let's see if
we can break down

00:00:31.214 --> 00:00:35.094
the 500x to the third into some
perfect squares and some things

00:00:35.094 --> 00:00:36.510
that maybe aren't
perfect squares.

00:00:36.510 --> 00:00:39.490
Because 500 is not a perfect
square, and x to the third

00:00:39.490 --> 00:00:43.140
is not a perfect square
right from the get go.

00:00:43.140 --> 00:00:45.970
So when in doubt, we can just
start to prime factorize this,

00:00:45.970 --> 00:00:48.070
and maybe we'll discover
some perfect squares.

00:00:48.070 --> 00:00:54.230
So 500 is divisible by
5, and it's 5 times 100.

00:00:54.230 --> 00:00:55.930
And you probably
already recognized

00:00:55.930 --> 00:00:57.710
that 100 is a perfect square.

00:00:57.710 --> 00:00:59.330
It is 10 times 10.

00:00:59.330 --> 00:01:02.410
But if you didn't know that,
you could keep factoring it.

00:01:02.410 --> 00:01:05.620
You could say, look,
100 is 2 times 50.

00:01:05.620 --> 00:01:11.710
50 is 2 times 25,
and 25 is 5 times 5.

00:01:11.710 --> 00:01:13.460
So you would see the
perfect squares here,

00:01:13.460 --> 00:01:17.220
because you would see two 2's.

00:01:17.220 --> 00:01:20.850
So you have a 2 times a 2,
and you have a 5 times a 5.

00:01:20.850 --> 00:01:23.450
So you could really just
say that 100 is 10 times 10,

00:01:23.450 --> 00:01:26.230
or you could say it's 2
squared times 5 squared.

00:01:26.230 --> 00:01:28.120
Either of those
things would allow

00:01:28.120 --> 00:01:30.300
you to break this
down a little bit.

00:01:30.300 --> 00:01:32.440
And then x to the
third, that's clearly

00:01:32.440 --> 00:01:40.001
just the same thing as
x squared times x, which

00:01:40.001 --> 00:01:42.000
allows us to break it
down into a perfect square

00:01:42.000 --> 00:01:44.230
and something that is
not a perfect square.

00:01:44.230 --> 00:01:45.930
So let's just
rewrite these numbers

00:01:45.930 --> 00:01:48.090
under the principal root sign.

00:01:48.090 --> 00:01:51.480
So this is going to
be equal to 3 times--

00:01:51.480 --> 00:01:54.960
and I'll save some
space here-- 3 times.

00:01:54.960 --> 00:01:59.980
The 500 I can rewrite
it as 100 times 5,

00:01:59.980 --> 00:02:03.200
and 100 I can rewrite
as 10 squared.

00:02:03.200 --> 00:02:05.159
So this right here I can
rewrite as 10 squared.

00:02:05.159 --> 00:02:07.241
I don't have to do all of
this business down here.

00:02:07.241 --> 00:02:09.800
But this will help you recognize
that it's a perfect square.

00:02:09.800 --> 00:02:13.720
It's 4 times 25, both of
those are perfect squares.

00:02:13.720 --> 00:02:15.480
So that's 10 squared.

00:02:15.480 --> 00:02:21.000
So I'm going to write this
as 10 squared times 5.

00:02:21.000 --> 00:02:23.880
That's the 500 right over there.

00:02:23.880 --> 00:02:30.370
And then I can rewrite the x to
the third as x squared times x.

00:02:32.890 --> 00:02:36.050
And then let me extend
my principal root sign

00:02:36.050 --> 00:02:37.380
right over here.

00:02:37.380 --> 00:02:39.740
And we know from basic
exponent properties,

00:02:39.740 --> 00:02:42.170
because that's really
what we're doing here,

00:02:42.170 --> 00:02:43.989
taking the square
root of something

00:02:43.989 --> 00:02:45.530
is really the same
thing as taking it

00:02:45.530 --> 00:02:46.910
to a fractional exponent.

00:02:46.910 --> 00:02:48.535
So it's the same
properties that apply.

00:02:48.535 --> 00:02:52.180
This is the same thing as
3 times the principal root

00:02:52.180 --> 00:02:57.990
of 10 squared times the
principal root-- let

00:02:57.990 --> 00:03:04.420
me do it this way-- times
the principal root of 5

00:03:04.420 --> 00:03:07.470
times the principal
root of x squared

00:03:07.470 --> 00:03:11.200
times the principal root of x.

00:03:11.200 --> 00:03:13.890
And now we can
simplify these things.

00:03:13.890 --> 00:03:16.270
So the principal
root of 10 squared,

00:03:16.270 --> 00:03:22.260
or the principal root of 100,
is going to be positive 10.

00:03:22.260 --> 00:03:23.729
Negative 10 is
also equal to 100,

00:03:23.729 --> 00:03:25.270
but when you take
the principal root,

00:03:25.270 --> 00:03:27.780
you only care about the
positive square root.

00:03:27.780 --> 00:03:31.460
So this right over
here is going to be 10.

00:03:31.460 --> 00:03:33.220
5 is not a perfect
square, so we'll just

00:03:33.220 --> 00:03:36.080
leave this expression
as the square root of 5.

00:03:36.080 --> 00:03:37.960
Square root of x squared.

00:03:37.960 --> 00:03:40.320
So here you might
be tempted, and I've

00:03:40.320 --> 00:03:41.700
seen this done many, many times.

00:03:41.700 --> 00:03:44.480
You're tempted to just say,
hey, that's just going to be x.

00:03:44.480 --> 00:03:46.590
The square root
of x squared is x.

00:03:46.590 --> 00:03:49.545
But what if what
if x is negative?

00:03:52.810 --> 00:03:55.440
Let's say that x was negative 2.

00:03:55.440 --> 00:03:59.420
Then x squared would be 4.

00:03:59.420 --> 00:04:03.390
And the principal root of
4 is equal to positive 2.

00:04:03.390 --> 00:04:05.600
It's equal to the
positive square root.

00:04:05.600 --> 00:04:07.780
So you can't just say
that this thing over here

00:04:07.780 --> 00:04:10.460
is going to be equal to
x, because in this case,

00:04:10.460 --> 00:04:12.770
it was equal to the
absolute value of x.

00:04:12.770 --> 00:04:14.100
And so that's what we do.

00:04:14.100 --> 00:04:17.420
We say that this is equal
to, regardless of whether x

00:04:17.420 --> 00:04:20.950
is positive or negative, we
want the positive square root.

00:04:20.950 --> 00:04:25.340
So you would say it's equal
to the absolute value of x.

00:04:25.340 --> 00:04:28.520
That's what that principal
root does for us.

00:04:28.520 --> 00:04:30.690
And so now, if we
simplify it a little bit,

00:04:30.690 --> 00:04:32.824
remember with multiplication,
since we're just

00:04:32.824 --> 00:04:34.490
multiplying a bunch
of expressions here,

00:04:34.490 --> 00:04:36.280
it doesn't matter
what order we do it.

00:04:36.280 --> 00:04:37.810
We can switch the order.

00:04:37.810 --> 00:04:39.840
So we have all the stuff
out of the radicals

00:04:39.840 --> 00:04:41.570
multiplying times
each other first,

00:04:41.570 --> 00:04:43.290
and then we have the radicals.

00:04:43.290 --> 00:04:46.716
So we have 3 times 10 times
the absolute value of x.

00:04:46.716 --> 00:04:50.140
3 times 10 is 30.

00:04:50.140 --> 00:04:52.800
So we have 30 times
the absolute value

00:04:52.800 --> 00:05:04.750
of x times the square root of
5 times the square root of x.

00:05:04.750 --> 00:05:07.300
And you have the principal
or the square root of 5

00:05:07.300 --> 00:05:10.070
and the square root of x, you
can actually rewrite this right

00:05:10.070 --> 00:05:13.377
here as the square root of 5x.

00:05:13.377 --> 00:05:15.210
You're taking the product
of two things that

00:05:15.210 --> 00:05:16.418
are raised to the same power.

00:05:16.418 --> 00:05:18.650
That's the same thing
as taking the product

00:05:18.650 --> 00:05:20.740
and then raising them to
that fractional power.

00:05:20.740 --> 00:05:22.323
Or I guess another
way to think of it,

00:05:22.323 --> 00:05:23.910
and then taking the square root.

00:05:23.910 --> 00:05:27.670
So this all
simplifies to 30 times

00:05:27.670 --> 00:05:34.725
the absolute value of x
times the square root of 5x.


Approximating square roots to hundredths

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.630
We are asked to approximate
the principal root,

00:00:03.630 --> 00:00:06.617
or the positive square root of
45, to the hundredths place.

00:00:06.617 --> 00:00:08.950
And I'm assuming they don't
want us to use a calculator.

00:00:08.950 --> 00:00:10.640
Because that would be too easy.

00:00:10.640 --> 00:00:12.950
So, let's see if we can
approximate this just

00:00:12.950 --> 00:00:15.640
with our pen and
paper right over here.

00:00:18.500 --> 00:00:21.880
So the square root of 45,
or the principal root of 45.

00:00:21.880 --> 00:00:27.370
45 is not a perfect square.

00:00:27.370 --> 00:00:29.437
It's definitely not
a perfect square.

00:00:29.437 --> 00:00:31.520
Let's see, what are the
perfect squares around it?

00:00:31.520 --> 00:00:33.650
We know that it is
going to be less

00:00:33.650 --> 00:00:36.990
than-- the next
perfect square above 45

00:00:36.990 --> 00:00:41.800
is going to be 49 because
that is 7 times 7--

00:00:41.800 --> 00:00:44.000
so it's less than
the square root of 49

00:00:44.000 --> 00:00:48.020
and it's greater than
the square root of 36.

00:00:48.020 --> 00:00:52.150
And so, the square root of
36, the principal root of 36

00:00:52.150 --> 00:00:53.930
I should say, is 6.

00:00:53.930 --> 00:00:59.450
And the principal
root of 49 is 7.

00:00:59.450 --> 00:01:06.430
So, this value right over here
is going to be between 6 and 7.

00:01:06.430 --> 00:01:10.130
And if we look at it, it's
only four away from 49.

00:01:10.130 --> 00:01:16.050
And it's nine away from 36.

00:01:16.050 --> 00:01:20.690
So, the different
between 36 and 49 is 13.

00:01:20.690 --> 00:01:25.330
So, it's a total 13 gap between
the 6 squared and 7 squared.

00:01:25.330 --> 00:01:28.410
And this is nine of
the way through it.

00:01:28.410 --> 00:01:30.864
So, just as a kind of
approximation maybe-- and it's

00:01:30.864 --> 00:01:33.280
not going to work out perfectly
because we're squaring it,

00:01:33.280 --> 00:01:35.450
this isn't a linear
relationship--

00:01:35.450 --> 00:01:39.780
but it's going to be closer to
7 than it's going to be to 6.

00:01:39.780 --> 00:01:43.880
At least the 45 is
9/13 of the way.

00:01:46.470 --> 00:01:46.970
Let's see.

00:01:46.970 --> 00:01:50.930
It looks like that's
about 2/3 of the way.

00:01:50.930 --> 00:01:58.980
So, let's try 6.7 as a guess
just based on 0.7 is about 2/3.

00:01:58.980 --> 00:02:00.450
It looks like about the same.

00:02:00.450 --> 00:02:03.259
Actually, we could calculate
this right here if we want.

00:02:03.259 --> 00:02:04.800
Actually, let's do
that just for fun.

00:02:04.800 --> 00:02:07.510
So 9/13 as a decimal
is going to be what?

00:02:07.510 --> 00:02:11.130
It's going to be 13 into 9.

00:02:11.130 --> 00:02:13.910
We're going to put some
decimal places right over here.

00:02:13.910 --> 00:02:16.980
13 doesn't go into 9
but 13 does go into 90.

00:02:16.980 --> 00:02:19.690
And it goes into 90-- let's
see, does it go into it

00:02:19.690 --> 00:02:24.160
seven times-- it goes
into it six times.

00:02:24.160 --> 00:02:28.910
So, 6 times 3 is 18.

00:02:28.910 --> 00:02:31.850
6 times 1 is 6, plus 1 is 7.

00:02:31.850 --> 00:02:34.620
And then you
subtract, you get 12.

00:02:34.620 --> 00:02:37.550
So, went into it almost
exactly seven times.

00:02:37.550 --> 00:02:40.490
So, this value right
here is almost a 0.7.

00:02:40.490 --> 00:02:43.500
And so if you say, how many
times does 13 go into 120?

00:02:43.500 --> 00:02:45.780
It looks like it's
like nine times?

00:02:45.780 --> 00:02:49.020
Yeah, it would go
into it nine times.

00:02:49.020 --> 00:02:50.384
9 times 3.

00:02:50.384 --> 00:02:51.050
Get rid of this.

00:02:51.050 --> 00:02:53.590
9 times 3 is 27.

00:02:53.590 --> 00:02:56.820
9 times 1 is 9, plus 2 is 11.

00:02:56.820 --> 00:02:59.340
You have a remainder of 3.

00:02:59.340 --> 00:03:01.900
It's about 0.69.

00:03:01.900 --> 00:03:04.860
So 6.7 would be a
pretty good guess.

00:03:04.860 --> 00:03:09.650
This is 0.69 of the
way between 36 and 49.

00:03:09.650 --> 00:03:13.520
So, let's go roughly 0.69
of the way between 6 and 7.

00:03:13.520 --> 00:03:15.270
So this is once again
just to approximate.

00:03:15.270 --> 00:03:17.561
It's not necessarily going
to give us the exact answer.

00:03:17.561 --> 00:03:20.100
We have to use that to
make a good initial guess.

00:03:20.100 --> 00:03:22.880
And then see how it works.

00:03:22.880 --> 00:03:27.580
Let's try 6.7.

00:03:27.580 --> 00:03:30.720
And the really way to
try it is to square 6.7.

00:03:30.720 --> 00:03:36.190
So 6.7 times-- maybe I'll
write the multiplication symbol

00:03:36.190 --> 00:03:40.080
there-- 6.7 times 6.7.

00:03:40.080 --> 00:03:43.230
So, we have 7 times 7 is 49.

00:03:43.230 --> 00:03:48.840
7 times 6 is 42, plus 4 is 46.

00:03:48.840 --> 00:03:54.520
Put a 0 now because we've
moved a space to the left.

00:03:54.520 --> 00:03:58.580
So, now we have 6 times 7 is 42.

00:03:58.580 --> 00:03:59.580
Carry the 4.

00:03:59.580 --> 00:04:03.730
6 times 6 is 36, plus 4 is 40.

00:04:03.730 --> 00:04:05.640
And so, 9 plus 0 is 9.

00:04:05.640 --> 00:04:06.640
6 plus 2 is 8.

00:04:06.640 --> 00:04:09.050
4 plus 0 is 4.

00:04:09.050 --> 00:04:10.710
And then we have a
4 right over here.

00:04:10.710 --> 00:04:13.001
And we have two total numbers
behind the decimal point.

00:04:13.001 --> 00:04:13.960
One, two.

00:04:13.960 --> 00:04:16.790
So this gives us 44.89.

00:04:16.790 --> 00:04:19.000
So, 6.7 gets us pretty close.

00:04:19.000 --> 00:04:21.615
But we're still not probably
right to the hundredth.

00:04:21.615 --> 00:04:23.740
Well, we're definitely not
to the hundredths place.

00:04:23.740 --> 00:04:25.680
This since we've only
gone to the tenths place

00:04:25.680 --> 00:04:26.810
right over here.

00:04:26.810 --> 00:04:35.020
So, if we want to get to 45, 6.7
squared is still less than 45,

00:04:35.020 --> 00:04:39.520
or 6.7 is still less than
the square root of 45.

00:04:39.520 --> 00:04:43.570
So let's try 6.71.

00:04:43.570 --> 00:04:45.790
Let me do this in a new color.

00:04:45.790 --> 00:04:47.940
I'll do 6.71 in pink.

00:04:47.940 --> 00:04:49.540
So, let's try 6.71.

00:04:49.540 --> 00:04:50.610
Increase it a little bit.

00:04:50.610 --> 00:04:52.720
See if we go from 44.89 to 45.

00:04:52.720 --> 00:04:55.544
Because this is
really close already.

00:04:55.544 --> 00:04:56.460
Let's just try it out.

00:04:56.460 --> 00:04:57.440
6.71.

00:04:57.440 --> 00:05:01.781
So once again, we have to
do some arithmetic by hand.

00:05:01.781 --> 00:05:03.280
We are assuming
that they don't want

00:05:03.280 --> 00:05:04.850
us to use a calculator here.

00:05:04.850 --> 00:05:06.510
So, we have 1 times 1 is 1.

00:05:06.510 --> 00:05:07.870
1 times 7 is 7.

00:05:07.870 --> 00:05:09.520
1 times 6 is 6.

00:05:09.520 --> 00:05:10.480
Put a 0 here.

00:05:10.480 --> 00:05:12.210
7 times 1 is 7.

00:05:12.210 --> 00:05:14.990
7 times 7 is 49.

00:05:14.990 --> 00:05:19.580
7 times 6 is 42, plus 4 is 46.

00:05:19.580 --> 00:05:21.140
And then we have two 0s here.

00:05:21.140 --> 00:05:22.800
6 times 1 is 6.

00:05:22.800 --> 00:05:26.320
6 times 7 is 42.

00:05:26.320 --> 00:05:27.810
Just have this new 4 here.

00:05:27.810 --> 00:05:33.280
6 times 6 is 36, plus 4 is 40.

00:05:33.280 --> 00:05:34.050
Plus 40.

00:05:36.745 --> 00:05:38.870
It's interesting to think
what we got incrementally

00:05:38.870 --> 00:05:43.274
by adding that one
hundredth over there.

00:05:43.274 --> 00:05:45.440
Well, we'll see actually
when we add all of this up.

00:05:45.440 --> 00:05:46.600
You get a 1.

00:05:46.600 --> 00:05:49.720
7 plus 7 is 14.

00:05:49.720 --> 00:05:56.300
1 plus 6 plus 9 is
16, plus 6 is 22.

00:05:56.300 --> 00:05:59.780
2 plus 6 plus 2 is 10.

00:05:59.780 --> 00:06:02.470
And then 1 plus 4 is 5.

00:06:02.470 --> 00:06:04.590
And then we bring down the 4.

00:06:04.590 --> 00:06:06.180
And we have one,
two, three, four

00:06:06.180 --> 00:06:07.554
numbers behind
the decimal point.

00:06:07.554 --> 00:06:10.040
One, two, three, four.

00:06:10.040 --> 00:06:13.050
So, when you we squared 6.71.

00:06:13.050 --> 00:06:19.300
6.71 squared is
equal to 45.0241.

00:06:19.300 --> 00:06:22.720
So 6.71 is a little bit greater.

00:06:22.720 --> 00:06:24.340
So, let me make it clear now.

00:06:24.340 --> 00:06:30.660
We know that 6.7 is less
than the square root of 45.

00:06:30.660 --> 00:06:33.764
And we know that
is less than 6.71.

00:06:33.764 --> 00:06:36.180
Because when we square this,
we get something a little bit

00:06:36.180 --> 00:06:38.220
over the square root of 45.

00:06:38.220 --> 00:06:45.490
But the key here is when we
square this, so 6.7 squared

00:06:45.490 --> 00:06:55.220
got us 44.89 which
is 0.11 away from 45.

00:06:55.220 --> 00:06:57.740
And then, if we look
at 6.71 squared,

00:06:57.740 --> 00:07:02.440
we're only 2.4
hundredths above 45.

00:07:02.440 --> 00:07:06.000
So, this right here is closer
to the square root of 45.

00:07:06.000 --> 00:07:08.000
So if we approximate to
the hundredths place,

00:07:08.000 --> 00:07:10.590
definitely want to go with 6.71.


Inverse variation application

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.690
We're told in this question
that on a string instrument,

00:00:03.690 --> 00:00:05.320
the length of a
string-- so let's

00:00:05.320 --> 00:00:08.550
call that l-- the
length of a string l

00:00:08.550 --> 00:00:12.220
varies inversely
as the frequency,

00:00:12.220 --> 00:00:16.219
so varies inversely
as the frequency.

00:00:16.219 --> 00:00:20.870
So l is going to be equal
to some constant times

00:00:20.870 --> 00:00:22.610
the inverse of the frequency.

00:00:25.635 --> 00:00:28.500
And I'll use f for
frequency-- the frequency

00:00:28.500 --> 00:00:29.460
of its vibrations.

00:00:29.460 --> 00:00:31.320
And then they tell
us the vibrations

00:00:31.320 --> 00:00:34.080
are what give the string
instruments their sound.

00:00:34.080 --> 00:00:35.480
That's nice.

00:00:35.480 --> 00:00:38.270
An 11-inch-- and it's
actually the vibrations

00:00:38.270 --> 00:00:40.490
of the string affecting
the air, and then

00:00:40.490 --> 00:00:42.880
the air compressions
eventually get to our eardrum.

00:00:42.880 --> 00:00:45.940
And that's actually what gives
us the perception of the sound.

00:00:45.940 --> 00:00:48.630
But we don't want to delve too
much into the physics of it.

00:00:48.630 --> 00:00:51.880
An 11-inch inches
string has a frequency.

00:00:51.880 --> 00:00:54.430
So 11-inch, so
this is its length.

00:00:54.430 --> 00:00:56.410
So the 11-inch string
has a frequency

00:00:56.410 --> 00:00:59.680
of 400 cycles per second.

00:00:59.680 --> 00:01:03.050
So this right here
is the frequency.

00:01:03.050 --> 00:01:05.470
A cycle per second is
also called a Hertz.

00:01:05.470 --> 00:01:07.670
Find the constant
of proportionality,

00:01:07.670 --> 00:01:11.060
and then find the frequency
of a 10-inch string.

00:01:11.060 --> 00:01:12.630
So they say an 11-inch string.

00:01:12.630 --> 00:01:18.390
So 11-inch string is
equal to some constant

00:01:18.390 --> 00:01:23.740
of proportionality times
1/400 cycles per second.

00:01:23.740 --> 00:01:30.870
So one over 400
cycles per second.

00:01:30.870 --> 00:01:32.570
I'll write second as sec.

00:01:32.570 --> 00:01:35.180
So to solve for the
constant of proportionality,

00:01:35.180 --> 00:01:38.690
we need to multiply both sides
by 400 cycles per second.

00:01:43.770 --> 00:01:50.660
Multiply the left hand side
by 400 cycles per second,

00:01:50.660 --> 00:01:54.710
and the left hand side
becomes 400 times 11.

00:01:54.710 --> 00:01:57.280
Well, 4 times 11 is 44.

00:01:57.280 --> 00:02:00.790
So 400 times 11 is 4,400.

00:02:00.790 --> 00:02:05.150
And then we have in our units,
just for out of interest,

00:02:05.150 --> 00:02:09.210
our units are cycles
times inches per second.

00:02:09.210 --> 00:02:13.780
So it's cycles times
inches in the numerator

00:02:13.780 --> 00:02:17.270
of our units divided by seconds.

00:02:17.270 --> 00:02:21.270
And that is equal to our
constant of proportionality.

00:02:21.270 --> 00:02:34.230
So we can say that the length is
equal to 4,400 times-- or 4,400

00:02:34.230 --> 00:02:39.520
cycles times inches per second--
I want to get the units right--

00:02:39.520 --> 00:02:43.620
per second, times 1
over the frequency.

00:02:47.330 --> 00:02:49.400
So we solved for our
constant of proportionality.

00:02:49.400 --> 00:02:51.530
And then we can use this
to find the frequency

00:02:51.530 --> 00:02:53.547
of a 10-inch string.

00:02:53.547 --> 00:02:55.130
So now we're talking
about a situation

00:02:55.130 --> 00:03:00.300
where our length is 10
inches, so 10 inches.

00:03:00.300 --> 00:03:07.170
So we get 10 inches
are equal to 4,400--

00:03:07.170 --> 00:03:10.770
as you can imagine, these
units are a little cumbersome--

00:03:10.770 --> 00:03:19.080
but 4,400 cycles times
inches per second times

00:03:19.080 --> 00:03:20.020
1 over the frequency.

00:03:23.547 --> 00:03:25.130
And so we could do
a couple of things.

00:03:25.130 --> 00:03:28.390
We could just multiply
both sides of this equation

00:03:28.390 --> 00:03:31.430
by the frequency so that it
gets out of the denominator.

00:03:31.430 --> 00:03:32.150
So let's do that.

00:03:32.150 --> 00:03:34.285
Let's multiply both
sides by the frequency.

00:03:37.000 --> 00:03:39.860
And then we could also divide
both sides by 10 inches

00:03:39.860 --> 00:03:40.670
to get rid of this.

00:03:40.670 --> 00:03:42.140
And then we'll just have
frequency on the left hand

00:03:42.140 --> 00:03:42.639
side.

00:03:42.639 --> 00:03:45.520
So we divide both
sides by 10 inches.

00:03:45.520 --> 00:03:47.255
And then we divide
it by 10 inches.

00:03:49.950 --> 00:03:54.000
The left hand side, we're
just left with the frequency.

00:03:54.000 --> 00:03:57.880
On the right hand side, we
have 4,400 divided by 10.

00:03:57.880 --> 00:04:03.310
4,400 divided by 10 is 440.

00:04:03.310 --> 00:04:07.420
And then you have cycles, inches
over second divided by inches.

00:04:07.420 --> 00:04:09.750
Well, the inches
cancel out with inches.

00:04:09.750 --> 00:04:14.450
And you're just left
with cycles per second.

00:04:14.450 --> 00:04:18.779
And then obviously, 1/f times
f cancels out to just being 1.

00:04:18.779 --> 00:04:21.450
So we get our frequency
when our string

00:04:21.450 --> 00:04:24.500
is when our string
is 10 inches long,

00:04:24.500 --> 00:04:27.960
our frequency has increased
to 440 cycles per second.

00:04:27.960 --> 00:04:30.510
From when it was 11 inches,
when it was slightly longer,

00:04:30.510 --> 00:04:32.920
our frequency was 400
cycles per second.

00:04:32.920 --> 00:04:35.350
When our string got little
bit shorter, one inch shorter,

00:04:35.350 --> 00:04:41.800
now our frequency increased
by 40 cycles per second.