Radical Equation Examples

WEBVTT
Kind: captions
Language: en

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Let's do a few more examples of
solving radical equations.

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Let's say I have the square root
of x plus 2 minus 2 is

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equal to 0.

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So just like I said in the last
video, we always want to

00:00:14.980 --> 00:00:16.170
isolate the radical.

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And the best way I can think
of doing it over here, is

00:00:18.890 --> 00:00:21.930
adding 2 to both sides
of this equation.

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So if you add 2 to both sides of
this equation, we are left

00:00:26.400 --> 00:00:31.390
with the square root of x plus
2-- on the left-hand side this

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cancels out --is equal to 2.

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And once you isolate one of
the radicals, then you're

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ready to essentially, square
both sides of this.

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If we're dealing with a
square root, we square

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both sides of this.

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Let's do that.

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So let's square both sides
of that equation.

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And then we get-- square root of
x plus 2 squared is just x

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plus 2 is equal to 2 squared.

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Is equal to 4.

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Subtract 2 from both sides
of this equation.

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We're left with x-- these 2's
cancel out --is equal to 2.

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Now let's validate that this
is truly the case.

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That it works when we're taking
the principal root, not

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both square roots or the
negative square root.

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Let's see, when we take the
principal root we get the

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square root of 2-- that's our
answer --plus this 2, minus 2,

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should be equal to 0.

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This is 4.

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The principal root of 4 is 2.

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2 minus 2 does indeed equal 0.

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So this is a valid solution.

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It's not an extraneous
solution.

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It works with the
principal root.

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Let's do one that doesn't
involve a square root.

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Let's do one with a
higher power root.

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Let's say we have the fourth
root of x squared minus 9 is

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equal to 2.

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So you say, gee, Sal,
what do I do here?

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Fourth root?

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That's crazy.

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How do we deal with
the fourth root?

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And it's the same process.

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When you had a square
root we squared it.

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When you have the fourth
root, we take it

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to the fourth power.

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And just as a bit of a review,
this could be completely

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written like this.

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Another way to write it would
be x squared minus 9 to the

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1/4 power is equal to 2.

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These are equivalent statements,
just different

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ways of writing the
fourth root.

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Now, if we raise both sides to
the fourth power, what happens

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to this exponent right here?

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If we raise something to an
exponent, and then raise that

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to another exponent, we multiply
the exponents.

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So 1/4 times 4 is 1.

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So this just becomes x squared
minus 9 to the first power.

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Or we could just write
x squared minus 9.

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And that's going to be equal
to 2 to the fourth power.

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2 to the fourth power
is, what?

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2 times 2 is 4.

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Times 2 is 8.

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Times 2 is 16.

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There's two ways we can do it
actually at this point.

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We can add 9 to both sides
of this equation.

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I'll do it this way in yellow.

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So if you add 9 you get x
squared is equal to 25.

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Take the square root, not just
the principal square root,

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take the positive and negative
square root of both sides of

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this equation and you get x is
equal to plus or minus 5.

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That's one way.

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The other way-- sometimes it's
confusing, people forget to

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take the positive and negative
square roots --is to do it the

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way we've been doing all along
through solving traditional

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quadratics, setting
them equal to 0.

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So what you do is you subtract
16 from both

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sides of this equation.

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If you do that you get x
squared minus 25, if we

00:03:40.180 --> 00:03:43.230
subtract 16 from both sides,
is equal to 0.

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So these two are equivalent.

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Essentially, we're just
moving the 25 onto

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the left-hand side.

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We're just subtracting
25 from both sides

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of this to get that.

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But what this does is it puts us
in the form of a difference

00:03:53.620 --> 00:03:54.370
of squares.

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We know we can rewrite this as
x plus 5 times x minus 5.

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We went over that several
videos ago.

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And that will be equal to 0.

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And if the product of two
numbers is equal to 0, that

00:04:08.120 --> 00:04:11.090
means that either one or both
of them is equal to 0.

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So x plus 5 is equal to 0 or
x minus 5 is equal to 0.

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And we get x is equal to
negative 5 or x is equal to

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positive 5, which is what
we got right there.

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Now, the one thing we want to
make sure once again, is that

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neither of these are extraneous
solutions.

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Because when you're taking a
fourth root or when you take

00:04:34.060 --> 00:04:35.520
something to the fourth
power, negative

00:04:35.520 --> 00:04:37.520
numbers also disappear.

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You can kind of view it as a
principal fourth root that

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we're taking here.

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A negative version of this could
have also been valid.

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So let's make sure that
they both work.

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Well, whether we put negative
5 or positive 5 here, this

00:04:49.630 --> 00:04:51.240
right here is going to be 25.

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So you get the fourth root of
25 minus 9, which is the

00:04:56.900 --> 00:04:59.930
fourth root of 16.

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So this is the fourth root of
16, which is equal to 16 to

00:05:04.160 --> 00:05:05.810
the 1/4 power.

00:05:05.810 --> 00:05:08.060
Which is indeed, 2.

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So both of these work.

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When you square either of these,
you do indeed, get 25.


Extraneous solutions to radical equations

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.710
In this video, we're going to
start having some experience

00:00:03.710 --> 00:00:08.400
solving radical equations or
equations that involve square

00:00:08.400 --> 00:00:11.610
roots or maybe even higher-power
roots, but we're

00:00:11.610 --> 00:00:14.680
going to also try to understand
an interesting

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phenomena that occurs when
we do these equations.

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Let me show you what
I'm talking about.

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Let's say I have the equation
the square root of x is equal

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to 2 times x minus 6.

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Now, one of the things you're
going to see whenever we do

00:00:29.710 --> 00:00:32.890
these radical equations is we
want to isolate at least one

00:00:32.890 --> 00:00:33.570
of the radicals.

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There's only one of them
in this equation.

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And when you isolate one of the
radicals on one side of

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the equation, this one starts
off like that, I have the

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square root of x isolated on
the left-hand side, then we

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square both sides
of the equation.

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So let us square both sides
of the equation.

00:00:48.750 --> 00:00:49.930
So I'll just rewrite it again.

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We'll do this one slowly.

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I'm going to square that and
that's going to be equal to 2x

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minus 6 squared.

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And squaring seems like
a valid operation.

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If that is equal to that, then
that squared should also be

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equal to that squared.

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So we just keep on going.

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So when you take the square root
of x and you square it,

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that'll just be x.

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And we get x is equal to-- this
squared is going to be 2x

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squared, which is 4x squared.

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It's 2x squared, the
whole thing.

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4x squared, and then you
multiply these two, which is

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negative 12x.

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And then it's going to be twice
that, so minus 24x.

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And then negative 6 squared
is plus 36.

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If you found going from this to
this difficult, you might

00:01:41.240 --> 00:01:44.750
want to review multiplying
polynomial expressions or

00:01:44.750 --> 00:01:47.020
multiplying binomials, or
actually, the special case

00:01:47.020 --> 00:01:48.070
where we square binomials.

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But the general view, it's this
squared, which is that.

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And then you have minus 2 times
the product of these 2.

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The product of those two is
minus 12x or negative 12x.

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2 times that is negative 24x,
and then that squared.

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So this is what our equation
has I guess we could say

00:02:03.340 --> 00:02:06.930
simplified to, and let's see
what happens if we subtract x

00:02:06.930 --> 00:02:08.710
from both sides of
this equation.

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So if you subtract x from both
sides of this equation, the

00:02:11.850 --> 00:02:14.710
left-hand side becomes zero
and the right-hand side

00:02:14.710 --> 00:02:21.990
becomes 4x squared minus
25x plus 36.

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So this radical equation his
simplified to just a standard

00:02:25.800 --> 00:02:27.670
quadratic equation.

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And just for simplicity, not
having to worry how to factor

00:02:30.080 --> 00:02:31.620
it and grouping and all of
that, let's just use the

00:02:31.620 --> 00:02:33.230
quadratic formula.

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So the quadratic formula tells
us that our solutions to this,

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x can be negative b.

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Negative 25.

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The negative of negative 25 is
positive 25 plus or minus the

00:02:45.540 --> 00:02:49.960
square root of 25 squared.

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25 squared is 625, minus 4 times
a, which is 4, times c,

00:02:59.200 --> 00:03:07.150
which is 36, all of that over 2
times 4, all of that over 8.

00:03:07.150 --> 00:03:08.870
So let's get our calculator
out to figure out what

00:03:08.870 --> 00:03:10.435
this is over here.

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So let's say so we have 625
minus-- let's see., this is

00:03:20.490 --> 00:03:23.470
going to be 16 times 36.

00:03:23.470 --> 00:03:28.280
16 times 36 is equal to 49.

00:03:28.280 --> 00:03:28.880
That's nice.

00:03:28.880 --> 00:03:29.920
It's a nice perfect square.

00:03:29.920 --> 00:03:31.830
We know what the square
root of 49 is.

00:03:31.830 --> 00:03:32.826
It's 7.

00:03:32.826 --> 00:03:34.780
So let me go back
to the problem.

00:03:34.780 --> 00:03:38.060
So this in here simplified
to 49.

00:03:38.060 --> 00:03:42.910
So x is equal to 25 plus or
minus the square root of 49,

00:03:42.910 --> 00:03:46.000
which is 7, all of
that over 8.

00:03:46.000 --> 00:03:50.960
So our two solutions here, if we
add 7, we get x is equal to

00:03:50.960 --> 00:03:57.870
25 plus 7 is 32, 32/8,
which is equal to 4.

00:03:57.870 --> 00:03:59.940
And then our other solution,
let me do that

00:03:59.940 --> 00:04:01.590
in a different color.

00:04:01.590 --> 00:04:07.340
x is equal to 25 minus
7, which is 18/8.

00:04:07.340 --> 00:04:12.680
8 goes into 18 two times,
remainder 2, so this is equal

00:04:12.680 --> 00:04:20.540
to 2 and 2/8 or 2 and 1/4
or 2.25, just like that.

00:04:20.540 --> 00:04:22.850
Now, I'm going to show
you an interesting

00:04:22.850 --> 00:04:24.220
phenomena that occurs.

00:04:24.220 --> 00:04:26.480
And maybe you might want to
pause it after I show you this

00:04:26.480 --> 00:04:28.080
conundrum, although I'm going
to tell you why this

00:04:28.080 --> 00:04:30.090
conundrum pops up.

00:04:30.090 --> 00:04:33.030
Let's try out to see if our
solutions actually work.

00:04:33.030 --> 00:04:35.560
So let's try x is equal to 4.

00:04:35.560 --> 00:04:38.680
If x is equal to 4 works, we
get the principal root of 4

00:04:38.680 --> 00:04:42.560
should be equal to 2
times 4 minus 6.

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The principal root of
4 is positive 2.

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Positive 2 should be equal to 2
times 4, which is 8 minus 6,

00:04:49.700 --> 00:04:50.420
which it is.

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This is true.

00:04:51.670 --> 00:04:54.290
So 4 works.

00:04:54.290 --> 00:04:57.420
Now, let's try to do
the same with 2.25.

00:04:57.420 --> 00:05:00.380
According to this, we should
be able to take the square

00:05:00.380 --> 00:05:05.593
root, the principal root of
2.2-- let me make my radical a

00:05:05.593 --> 00:05:06.470
little bit bigger.

00:05:06.470 --> 00:05:10.840
The principal root of 2.25
should be equal to 2 times

00:05:10.840 --> 00:05:15.620
2.25 minus 6.

00:05:15.620 --> 00:05:18.420
Now, you may or may not be able
to do this in your head.

00:05:18.420 --> 00:05:22.520
You might know that the square
root of 225 is 15.

00:05:22.520 --> 00:05:25.180
And then from that, you might
be able to figure out the

00:05:25.180 --> 00:05:29.180
square root of 2.25 is 1.5.

00:05:29.180 --> 00:05:32.970
Let me just use the calculator
to verify that for you.

00:05:32.970 --> 00:05:35.410
So 2.25, take the square root.

00:05:35.410 --> 00:05:36.600
It's 1.5.

00:05:36.600 --> 00:05:39.170
The principal root is 1.5.

00:05:39.170 --> 00:05:40.770
Another square root
is negative 1.5.

00:05:40.770 --> 00:05:42.020
So it's 1.5.

00:05:44.160 --> 00:05:45.980
And then, according to this,
this should be equal to 2

00:05:45.980 --> 00:05:50.990
times 2.25 is 4.5 minus 6.

00:05:50.990 --> 00:05:52.970
Now, is this true?

00:05:52.970 --> 00:05:57.350
This is telling us that 1.5
is equal to negative 1.5.

00:05:57.350 --> 00:05:59.200
This is not true.

00:05:59.200 --> 00:06:03.790
2.5 did not work for this
radical equation.

00:06:03.790 --> 00:06:06.580
We call this an extraneous
solution.

00:06:06.580 --> 00:06:13.730
So 2.25 is an extraneous
solution.

00:06:13.730 --> 00:06:18.070
Now, here's the conundrum: Why
did we get 2.25 as an answer?

00:06:18.070 --> 00:06:20.310
It looks like we did very valid
things the whole way

00:06:20.310 --> 00:06:23.830
down, and we got a quadratic,
and we got 2.25.

00:06:23.830 --> 00:06:25.260
And there's a hint here.

00:06:25.260 --> 00:06:28.010
When we substitute 2.25,
we get 1.5 is

00:06:28.010 --> 00:06:30.550
equal to negative 1.5.

00:06:30.550 --> 00:06:34.120
So there's something here,
something we did gave us this

00:06:34.120 --> 00:06:37.430
solution that doesn't quite
apply over here.

00:06:37.430 --> 00:06:38.720
And I'll give you
another hint.

00:06:38.720 --> 00:06:40.090
Let's try it at this step.

00:06:40.090 --> 00:06:42.320
If you look at this step, you're
going to see that both

00:06:42.320 --> 00:06:44.300
solutions actually work.

00:06:44.300 --> 00:06:45.990
So you could try it
out if you like.

00:06:45.990 --> 00:06:47.740
Actually, try it out
on your own time.

00:06:47.740 --> 00:06:49.865
Put in 2.25 for x here.

00:06:49.865 --> 00:06:51.520
You're going to see
that it works.

00:06:51.520 --> 00:06:55.060
Put in 4 for x here and you see
that they both work here.

00:06:55.060 --> 00:06:57.360
So they're both valid
solutions to that.

00:07:00.400 --> 00:07:05.330
So something happened when we
squared that made the equation

00:07:05.330 --> 00:07:07.070
a little bit different.

00:07:07.070 --> 00:07:10.000
There's something slightly
different about this equation

00:07:10.000 --> 00:07:11.550
than that equation.

00:07:11.550 --> 00:07:15.710
And the answer is there's two
ways you could think about it.

00:07:15.710 --> 00:07:20.430
To go back from this equation to
that equation, we take the

00:07:20.430 --> 00:07:21.440
square root.

00:07:21.440 --> 00:07:24.260
But to be more particular about
it, we are taking the

00:07:24.260 --> 00:07:27.020
principal root of both sides.

00:07:27.020 --> 00:07:30.410
Now, you could take the negative
square root as well.

00:07:30.410 --> 00:07:33.710
Notice, this is only taking
the principal square root.

00:07:33.710 --> 00:07:36.430
Going from this right here--
let me be very clear.

00:07:36.430 --> 00:07:40.930
This statement, we already
established that both of these

00:07:40.930 --> 00:07:43.610
solutions, both the valid
solution and the extraneous

00:07:43.610 --> 00:07:45.350
solution to this radical
equation,

00:07:45.350 --> 00:07:47.060
satisfy this right here.

00:07:47.060 --> 00:07:49.970
Only the valid one satisfies
the original problem.

00:07:49.970 --> 00:07:52.390
So let me write the equation
that both of them satisfy.

00:07:52.390 --> 00:07:55.550
Because this is really an
interesting conundrum.

00:07:55.550 --> 00:07:58.370
And I think it gives you a
little bit of a nuance and

00:07:58.370 --> 00:07:59.860
kind of tells you what's
happening when we take

00:07:59.860 --> 00:08:01.320
principal roots of things.

00:08:01.320 --> 00:08:04.230
And why when you square both
sides, you are, to some

00:08:04.230 --> 00:08:06.660
degree, you could either think
of it as losing or gaining

00:08:06.660 --> 00:08:07.960
some information.

00:08:07.960 --> 00:08:11.720
Now, this could be written
as x is equal

00:08:11.720 --> 00:08:16.450
to 2x minus 6 squared.

00:08:16.450 --> 00:08:19.090
This is one valid interpretation
of this

00:08:19.090 --> 00:08:20.470
equation right here.

00:08:20.470 --> 00:08:24.290
But there's a completely other
legitimate interpretation of

00:08:24.290 --> 00:08:27.220
this equation.

00:08:27.220 --> 00:08:31.300
This could also be x is
equal to negative 1

00:08:31.300 --> 00:08:35.400
times 2x minus 6 squared.

00:08:35.400 --> 00:08:37.870
And why are these equal
interpretations?

00:08:37.870 --> 00:08:41.010
Because when you square the
negative 1, the negative 1

00:08:41.010 --> 00:08:42.610
will disappear.

00:08:42.610 --> 00:08:44.420
These are equivalent
statements.

00:08:44.420 --> 00:08:47.500
And another way of writing
this one, another way of

00:08:47.500 --> 00:08:50.630
writing this right here, is
that x is equal to-- you

00:08:50.630 --> 00:08:52.450
multiply negative
1 times that.

00:08:52.450 --> 00:08:58.080
You get negative 2x plus 6
or 6 minus 2x squared.

00:08:58.080 --> 00:09:04.560
This and this are two ways
of writing that.

00:09:04.560 --> 00:09:07.760
Now, when we took our square
root or when we-- I guess

00:09:07.760 --> 00:09:09.700
there's two ways you
can think about it.

00:09:09.700 --> 00:09:13.170
When we squared it, we're
assuming that this was the

00:09:13.170 --> 00:09:15.800
only interpretation, but
this was the other one.

00:09:15.800 --> 00:09:21.460
So we found two solutions to
this, but only 4 satisfies

00:09:21.460 --> 00:09:24.060
this interpretation
right here.

00:09:24.060 --> 00:09:26.800
I hope you get what I'm saying
because we're kind of only

00:09:26.800 --> 00:09:29.710
taking-- you can imagine the
positive square root.

00:09:29.710 --> 00:09:32.300
We're not considering the
negative square root of this,

00:09:32.300 --> 00:09:34.080
because when you take the square
root of both sides to

00:09:34.080 --> 00:09:36.010
get here, we're only taking
the principal root.

00:09:36.010 --> 00:09:37.880
Another way to view it--
let me rewrite

00:09:37.880 --> 00:09:39.130
the original equation.

00:09:42.560 --> 00:09:48.630
We had the square root of x
is equal to 2x minus 6.

00:09:48.630 --> 00:09:50.620
Now, we said 4 is a solution.

00:09:50.620 --> 00:09:52.790
2.25 isn't a solution.

00:09:52.790 --> 00:09:56.690
2.25 would've been a solution if
we said both of the square

00:09:56.690 --> 00:09:59.800
roots of x is equal
to 2x minus 6.

00:09:59.800 --> 00:10:04.220
Now you try it out and
2.25 will have a

00:10:04.220 --> 00:10:05.330
valid solution here.

00:10:05.330 --> 00:10:10.050
If you take the negative square
root of 2.25, that is

00:10:10.050 --> 00:10:15.590
equal to 2 times 2.25, so that
is equal to 4.5 minus 6, which

00:10:15.590 --> 00:10:17.230
is negative 1.5.

00:10:17.230 --> 00:10:18.430
That is true.

00:10:18.430 --> 00:10:21.200
The positive version is where
you get x is equal to 4.

00:10:21.200 --> 00:10:23.250
So that's why we got
two solutions.

00:10:23.250 --> 00:10:25.900
And if you square this-- maybe
this is an easier way to

00:10:25.900 --> 00:10:26.300
remember it.

00:10:26.300 --> 00:10:31.780
If you square this, you actually
get this equation

00:10:31.780 --> 00:10:34.150
that both solutions are valid.

00:10:34.150 --> 00:10:35.900
Now, you might have found
that a little bit

00:10:35.900 --> 00:10:37.070
confusing and all of that.

00:10:37.070 --> 00:10:38.990
My intention is not
to confuse you.

00:10:38.990 --> 00:10:41.490
The simple thing to think about
when you are solving

00:10:41.490 --> 00:10:45.060
radical equations is, look,
isolate radicals, square, keep

00:10:45.060 --> 00:10:45.740
on solving.

00:10:45.740 --> 00:10:47.460
You might get more
than one answer.

00:10:47.460 --> 00:10:49.180
Plug your answers back in.

00:10:49.180 --> 00:10:52.170
Answers that don't work, they're
extraneous solutions.

00:10:52.170 --> 00:10:54.260
But most of my explanation in
this video is really why does

00:10:54.260 --> 00:10:56.290
that extraneous solution
pop up?

00:10:56.290 --> 00:11:00.900
And hopefully, I gave you some
intuition that our equation is

00:11:00.900 --> 00:11:02.610
the square root of x.

00:11:02.610 --> 00:11:05.140
The extraneous solution would
be valid if we took the plus

00:11:05.140 --> 00:11:09.010
or minus square root of x, not
just the principal root.


How to rationalize a denominator

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:03.160
In this video, we're going to
learn how to rationalize the

00:00:03.160 --> 00:00:04.410
denominator.

00:00:08.440 --> 00:00:11.610
What we mean by that is, let's
say we have a fraction that

00:00:11.610 --> 00:00:15.570
has a non-rational denominator,
the simplest one

00:00:15.570 --> 00:00:19.400
I can think of is 1 over
the square root of 2.

00:00:19.400 --> 00:00:21.440
So to rationalize this
denominator, we're going to

00:00:21.440 --> 00:00:24.910
just re-represent this number in
some way that does not have

00:00:24.910 --> 00:00:27.630
an irrational number
in the denominator.

00:00:27.630 --> 00:00:29.350
Now the first question
you might ask is,

00:00:29.350 --> 00:00:31.710
Sal, why do we care?

00:00:31.710 --> 00:00:34.160
Why must we rationalize
denominators?

00:00:34.160 --> 00:00:36.030
And you don't have to
rationalize them.

00:00:36.030 --> 00:00:40.260
But I think the reason why
this is in many algebra

00:00:40.260 --> 00:00:43.230
classes and why many teachers
want you to, is it gets the

00:00:43.230 --> 00:00:45.270
numbers into a common format.

00:00:45.270 --> 00:00:48.150
And I also think, I've been
told that back in the day

00:00:48.150 --> 00:00:52.360
before we had calculators that
some forms of computation,

00:00:52.360 --> 00:00:55.170
people found it easier to have
a rational number in the

00:00:55.170 --> 00:00:55.600
denominator.

00:00:55.600 --> 00:00:57.200
I don't know if that's
true or not.

00:00:57.200 --> 00:01:00.070
And then, the other reason
is just for aesthetics.

00:01:00.070 --> 00:01:02.430
Some people say, I don't
like saying what 1

00:01:02.430 --> 00:01:03.640
square root of 2 is.

00:01:03.640 --> 00:01:04.180
I don't even know.

00:01:04.180 --> 00:01:05.900
You know, I want to how
big the pie is.

00:01:05.900 --> 00:01:08.330
I want a denominator to
be a rational number.

00:01:08.330 --> 00:01:11.420
So with that said, let's learn
how to rationalize it.

00:01:11.420 --> 00:01:15.260
So the simple way, if you just
have a simple irrational

00:01:15.260 --> 00:01:17.030
number in the denominator just
like that, you can just

00:01:17.030 --> 00:01:21.490
multiply the numerator and the
denominator by that irrational

00:01:21.490 --> 00:01:24.100
number over that irrational
number.

00:01:24.100 --> 00:01:25.670
Now this is clearly just 1.

00:01:25.670 --> 00:01:28.610
Anything over anything or
anything over that same thing

00:01:28.610 --> 00:01:29.410
is going to be 1.

00:01:29.410 --> 00:01:31.480
So we're not fundamentally
changing the number.

00:01:31.480 --> 00:01:33.610
We're just changing how
we represent it.

00:01:33.610 --> 00:01:35.470
So what's this going
to be equal to?

00:01:35.470 --> 00:01:38.170
The numerator is going to be 1
times the square root of 2,

00:01:38.170 --> 00:01:39.620
which is the square root of 2.

00:01:39.620 --> 00:01:43.230
The denominator is going to be
the square root of 2 times the

00:01:43.230 --> 00:01:44.640
square root of 2.

00:01:44.640 --> 00:01:46.180
Well the square root of
2 times the square

00:01:46.180 --> 00:01:48.130
root of 2 is 2.

00:01:48.130 --> 00:01:50.460
That is 2.

00:01:50.460 --> 00:01:54.180
By definition, this squared
must be equal to 2.

00:01:54.180 --> 00:01:55.030
And we are squaring it.

00:01:55.030 --> 00:01:56.400
We're multiplying
it by itself.

00:01:56.400 --> 00:01:57.630
So that is equal to 2.

00:01:57.630 --> 00:01:59.690
We have rationalized
the denominator.

00:01:59.690 --> 00:02:02.950
We haven't gotten rid of the
radical sign, but we've

00:02:02.950 --> 00:02:03.680
brought it to the numerator.

00:02:03.680 --> 00:02:10.120
And now in the denominator we
have a rational number.

00:02:10.120 --> 00:02:13.610
And you could say, hey, now I
have square root of 2 halves.

00:02:13.610 --> 00:02:15.860
It's easier to say even,
so maybe that's another

00:02:15.860 --> 00:02:18.370
justification for rationalizing
this

00:02:18.370 --> 00:02:19.840
denominator.

00:02:19.840 --> 00:02:22.040
Let's do a couple
more examples.

00:02:22.040 --> 00:02:27.750
Let's say I had 7 over the
square root of 15.

00:02:27.750 --> 00:02:30.320
So the first thing I'd want to
do is just simplify this

00:02:30.320 --> 00:02:31.930
radical right here.

00:02:31.930 --> 00:02:32.250
Let's see.

00:02:32.250 --> 00:02:32.920
Square root of 15.

00:02:32.920 --> 00:02:34.530
15 is 3 times 5.

00:02:34.530 --> 00:02:36.570
Neither of those are
perfect squares.

00:02:36.570 --> 00:02:39.310
So actually, this is about as
simple as I'm going to get.

00:02:39.310 --> 00:02:43.770
So just like we did here, let's
multiply this times the

00:02:43.770 --> 00:02:48.490
square root of 15 over the
square root of 15.

00:02:48.490 --> 00:02:52.720
And so this is going to be equal
to 7 times the square

00:02:52.720 --> 00:02:53.720
root of 15.

00:02:53.720 --> 00:02:55.590
Just multiply the numerators.

00:02:55.590 --> 00:02:58.690
Over square root of 15 times
the square root of 15.

00:02:58.690 --> 00:03:00.590
That's 15.

00:03:00.590 --> 00:03:03.020
So once again, we have
rationalized the denominator.

00:03:03.020 --> 00:03:05.350
This is now a rational number.

00:03:05.350 --> 00:03:07.840
We essentially got the radical
up on the top or we got the

00:03:07.840 --> 00:03:10.510
irrational number up
on the numerator.

00:03:10.510 --> 00:03:13.190
We haven't changed the number,
we just changed how we are

00:03:13.190 --> 00:03:14.700
representing it.

00:03:14.700 --> 00:03:17.920
Now, let's take it up
one more level.

00:03:17.920 --> 00:03:24.900
What happens if we have
something like 12 over 2 minus

00:03:24.900 --> 00:03:27.080
the square root of 5?

00:03:27.080 --> 00:03:29.940
So in this situation, we
actually have a binomial in

00:03:29.940 --> 00:03:30.460
the denominator.

00:03:30.460 --> 00:03:33.950
And this binomial contains
an irrational number.

00:03:33.950 --> 00:03:35.000
I can't do the trick here.

00:03:35.000 --> 00:03:37.230
If I multiplied this by square
root of 5 over square root of

00:03:37.230 --> 00:03:39.380
5, I'm still going to have an
irrational denominator.

00:03:39.380 --> 00:03:40.460
Let me just show you.

00:03:40.460 --> 00:03:42.210
Just to show you
it won't work.

00:03:42.210 --> 00:03:44.720
If I multiplied this square root
of 5 over square root of

00:03:44.720 --> 00:03:47.730
5, the numerator is going
to be 12 times the

00:03:47.730 --> 00:03:49.300
square root of 5.

00:03:49.300 --> 00:03:51.170
The denominator, we have
to distribute this.

00:03:51.170 --> 00:03:54.670
It's going to be 2 times the
square root of 5 minus the

00:03:54.670 --> 00:03:57.820
square root of 5 times the
square root of 5, which is 5.

00:03:57.820 --> 00:04:00.180
So you see, in this situation,
it didn't help us.

00:04:00.180 --> 00:04:02.570
Because the square root of 5,
although this part became

00:04:02.570 --> 00:04:06.110
rational,it became a 5, this
part became irrational.

00:04:06.110 --> 00:04:07.900
2 times the square root of 5.

00:04:07.900 --> 00:04:11.370
So this is not what you want
to do where you have a

00:04:11.370 --> 00:04:14.120
binomial that contains an
irrational number in the

00:04:14.120 --> 00:04:14.780
denominator.

00:04:14.780 --> 00:04:18.019
What you do here is use our
skills when it comes to

00:04:18.019 --> 00:04:19.170
difference of squares.

00:04:19.170 --> 00:04:21.209
So let's just take a
little side here.

00:04:21.209 --> 00:04:25.470
We learned a long time ago--
well, not that long ago.

00:04:25.470 --> 00:04:29.570
If you had 2 minus the square
root of 5 and you multiply

00:04:29.570 --> 00:04:34.900
that by 2 plus the square root
of 5, what will this get you?

00:04:34.900 --> 00:04:36.670
Now you might remember.

00:04:36.670 --> 00:04:38.630
And if you don't recognize this
immediately, this is the

00:04:38.630 --> 00:04:43.790
exact same pattern as a minus
b times a plus b.

00:04:48.290 --> 00:04:51.040
Which we've seen several
videos ago is a

00:04:51.040 --> 00:04:53.770
squared minus b squared.

00:04:53.770 --> 00:04:54.470
Little bit of review.

00:04:54.470 --> 00:04:56.900
This is a times a, which
is a squared. a

00:04:56.900 --> 00:04:59.430
times b, which is ab.

00:04:59.430 --> 00:05:02.630
Minus b times a, which
is minus ab.

00:05:02.630 --> 00:05:04.840
And then, negative b
times a positive

00:05:04.840 --> 00:05:07.150
b, negative b squared.

00:05:07.150 --> 00:05:09.070
These cancel out and you're
just left with a

00:05:09.070 --> 00:05:10.980
squared minus b squared.

00:05:10.980 --> 00:05:15.710
So 2 minus the square root of 5
times 2 plus the square root

00:05:15.710 --> 00:05:22.480
of 5 is going to be equal to
2 squared, which is 4.

00:05:22.480 --> 00:05:23.260
Let me write it that way.

00:05:23.260 --> 00:05:26.920
It's going to be equal to 2
squared minus the square root

00:05:26.920 --> 00:05:30.690
of 5 squared, which is just 5.

00:05:30.690 --> 00:05:34.345
So this would just be equal to
4 minus 5 or negative 1.

00:05:36.920 --> 00:05:39.520
If you take advantage of the
difference of squares of

00:05:39.520 --> 00:05:43.080
binomials, or the factoring
difference of squares, however

00:05:43.080 --> 00:05:45.920
you want to view it, then
you can rationalize this

00:05:45.920 --> 00:05:46.240
denominator.

00:05:46.240 --> 00:05:48.840
So let's do that.

00:05:48.840 --> 00:05:49.750
Let me rewrite the problem.

00:05:49.750 --> 00:05:53.280
12 over 2 minus the
square root of 5.

00:05:53.280 --> 00:05:56.810
In this situation, I just
multiply the numerator and the

00:05:56.810 --> 00:06:02.270
denominator by 2 plus the square
root of 5 over 2 plus

00:06:02.270 --> 00:06:03.600
the square root of 5.

00:06:03.600 --> 00:06:05.580
Once again, I'm just multiplying
the number by 1.

00:06:05.580 --> 00:06:07.960
So I'm not changing the
fundamental number.

00:06:07.960 --> 00:06:10.020
I'm just changing how
we represent it.

00:06:10.020 --> 00:06:11.780
So the numerator is
going to become 12

00:06:11.780 --> 00:06:14.010
times 2, which is 24.

00:06:14.010 --> 00:06:16.125
Plus 12 times the square
root of 5.

00:06:22.640 --> 00:06:25.470
Once again, this is
like a factored

00:06:25.470 --> 00:06:27.560
difference of squares.

00:06:27.560 --> 00:06:30.200
This is going to be equal to 2
squared, which is going to be

00:06:30.200 --> 00:06:31.450
exactly equal to that.

00:06:31.450 --> 00:06:35.330
Which is 4 minus 1, or we
could just-- sorry.

00:06:35.330 --> 00:06:38.040
4 minus 5.

00:06:38.040 --> 00:06:40.750
It's 2 squared minus square
root of 5 squared.

00:06:40.750 --> 00:06:42.610
So it's 4 minus 5.

00:06:42.610 --> 00:06:46.720
Or we could just write that
as minus 1, or negative 1.

00:06:46.720 --> 00:06:47.900
Or we could put a 1
there and put a

00:06:47.900 --> 00:06:49.590
negative sign out in front.

00:06:49.590 --> 00:06:51.720
And then, no point in even
putting a 1 in the

00:06:51.720 --> 00:06:52.170
denominator.

00:06:52.170 --> 00:06:57.120
We could just say that this is
equal to negative 24 minus 12

00:06:57.120 --> 00:06:59.140
square roots of 5.

00:06:59.140 --> 00:07:01.250
So this case, it kind of did
simplify it as well.

00:07:01.250 --> 00:07:02.930
It wasn't just for the sake
of rationalizing it.

00:07:02.930 --> 00:07:04.770
It actually made it look
a little bit better.

00:07:04.770 --> 00:07:06.490
And you know, I don't if I
mentioned in the beginning,

00:07:06.490 --> 00:07:09.540
this is good because
it's not obvious.

00:07:09.540 --> 00:07:12.190
If you and I are both trying to
build a rocket and you get

00:07:12.190 --> 00:07:16.200
this as your answer and I get
this as my answer, this isn't

00:07:16.200 --> 00:07:18.260
obvious, at least to me just by
looking at it, that they're

00:07:18.260 --> 00:07:19.500
the same number.

00:07:19.500 --> 00:07:22.450
But if we agree to always
rationalize our denominators,

00:07:22.450 --> 00:07:23.380
we're like, oh great.

00:07:23.380 --> 00:07:24.450
We got the same number.

00:07:24.450 --> 00:07:27.320
Now we're ready to send
our rocket to Mars.

00:07:27.320 --> 00:07:30.910
Let's do one more of this, one
more of these right here.

00:07:39.780 --> 00:07:41.900
Let's do one with
variables in it.

00:07:41.900 --> 00:07:48.060
So let's say we have 5y over
2 times the square

00:07:48.060 --> 00:07:52.700
root of y minus 5.

00:07:52.700 --> 00:07:54.530
So we're going to do this
exact same process.

00:07:54.530 --> 00:07:57.640
We have a binomial with an
irrational denominator.

00:07:57.640 --> 00:07:58.690
It might be a rational.

00:07:58.690 --> 00:07:59.920
We don't know what y is.

00:07:59.920 --> 00:08:02.020
But y can take on any value, so
at points it's going to be

00:08:02.020 --> 00:08:02.570
irrational.

00:08:02.570 --> 00:08:05.270
So we really just don't want a
radical in the denominator.

00:08:05.270 --> 00:08:07.050
So what is this going
to be equal to?

00:08:07.050 --> 00:08:09.180
Well, let's just multiply the
numerator and the denominator

00:08:09.180 --> 00:08:15.780
by 2 square roots of y
plus 5 over 2 square

00:08:15.780 --> 00:08:19.100
roots of y plus 5.

00:08:19.100 --> 00:08:20.150
This is just 1.

00:08:20.150 --> 00:08:21.690
We are not changing the
number, we're just

00:08:21.690 --> 00:08:23.860
multiplying it by 1.

00:08:23.860 --> 00:08:25.260
So let's start with
the denominator.

00:08:25.260 --> 00:08:27.570
What is the denominator
going to be equal to?

00:08:27.570 --> 00:08:30.940
The denominator is going to
be equal to this squared.

00:08:30.940 --> 00:08:32.409
Once again, just a difference
of squares.

00:08:32.409 --> 00:08:35.880
It's going to be 2 times
the square root of y

00:08:35.880 --> 00:08:40.830
squared minus 5 squared.

00:08:40.830 --> 00:08:44.630
If you factor this, you would
get 2 square roots of y plus 5

00:08:44.630 --> 00:08:46.630
times 2 square roots
of y minus 5.

00:08:46.630 --> 00:08:48.440
This is a difference
of squares.

00:08:48.440 --> 00:08:52.770
And then our numerator is 5y
times 2 square roots of y.

00:08:52.770 --> 00:08:56.360
So it would be 10.

00:08:56.360 --> 00:08:58.330
And this is y to the first
power, this is

00:08:58.330 --> 00:09:00.080
y to the half power.

00:09:00.080 --> 00:09:02.370
We could write y square
roots of y.

00:09:02.370 --> 00:09:04.430
10y square roots of y.

00:09:04.430 --> 00:09:07.380
Or we could write this as y to
the 3/2 power or 1 and 1/2

00:09:07.380 --> 00:09:09.300
power, however you
want to view it.

00:09:09.300 --> 00:09:15.820
And then finally, 5y times
5 is plus 25y.

00:09:15.820 --> 00:09:18.550
And we can simplify
this further.

00:09:18.550 --> 00:09:21.250
So what is our denominator
going to be equal to?

00:09:21.250 --> 00:09:24.490
We're going to have 2
squared, which is 4.

00:09:24.490 --> 00:09:29.410
Square root of y squared is y.

00:09:29.410 --> 00:09:30.800
4y.

00:09:30.800 --> 00:09:34.870
And then minus 25.

00:09:34.870 --> 00:09:40.140
And our numerator over here is--
We could even write this.

00:09:40.140 --> 00:09:42.690
We could keep it exactly the
way we've written it here.

00:09:42.690 --> 00:09:43.600
We could factor out a y.

00:09:43.600 --> 00:09:44.840
There's all sorts of things
we could do it.

00:09:44.840 --> 00:09:48.540
But just to keep things simple,
we could just leave

00:09:48.540 --> 00:09:50.340
that as 10.

00:09:50.340 --> 00:09:51.100
Let me just write
it different.

00:09:51.100 --> 00:09:53.520
I could write that as this is
y to the first, this is y to

00:09:53.520 --> 00:09:54.970
the 1/2 power.

00:09:54.970 --> 00:09:58.200
I could write that as even
a y to the 3/2 if I want.

00:09:58.200 --> 00:10:02.240
I could write that as y to
the 1 and 1/2 if I want.

00:10:02.240 --> 00:10:05.650
Or I could write that as 10y
times the square root of y.

00:10:05.650 --> 00:10:06.900
All of those are equivalent.

00:10:10.010 --> 00:10:12.080
Plus 25y.

00:10:12.080 --> 00:10:14.990
Anyway, hopefully you found
this rationalizing the

00:10:14.990 --> 00:10:17.110
denominator interesting.


Adding and simplifying radicals

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.900
In this video, I'm going to
do some more examples of

00:00:02.900 --> 00:00:04.880
simplifying radical
expressions.

00:00:04.880 --> 00:00:06.990
But these are going to involve
adding and subtracting

00:00:06.990 --> 00:00:08.200
different radical expressions.

00:00:08.200 --> 00:00:10.850
And I think it's a good tool
to have in your toolkit in

00:00:10.850 --> 00:00:12.080
case you've never
seen it before.

00:00:12.080 --> 00:00:13.520
So let's do a few of these.

00:00:13.520 --> 00:00:17.710
So let's say I have 3 times
the square root of 8-- we

00:00:17.710 --> 00:00:19.836
learned before that's actually
the principal square root of

00:00:19.836 --> 00:00:24.010
8, or the positive square root
of 8-- minus 6 times the

00:00:24.010 --> 00:00:27.025
principal square root of 32.

00:00:27.025 --> 00:00:29.950
So let's see what we can
do to simplify this.

00:00:29.950 --> 00:00:35.110
So first of all, 8, we can
write that as 2 times 4.

00:00:35.110 --> 00:00:36.605
When 4 is a perfect
square, you might

00:00:36.605 --> 00:00:37.400
already recognize that.

00:00:37.400 --> 00:00:39.990
We could further factor
that into 2 times 2.

00:00:39.990 --> 00:00:40.940
But I don't think we need to.

00:00:40.940 --> 00:00:45.770
So we can rewrite 3 square
root of 8 as 3 times the

00:00:45.770 --> 00:00:50.430
square root of 4 times
the square root of 2.

00:00:50.430 --> 00:00:52.830
This is the same thing as the
square root of 4 times 2,

00:00:52.830 --> 00:00:54.490
which is the square root of 8.

00:00:54.490 --> 00:00:57.220
So this term is the same
thing as that term.

00:00:57.220 --> 00:00:58.890
And then, let's look at 32.

00:00:58.890 --> 00:01:01.050
We want to do the square
root of 32.

00:01:01.050 --> 00:01:04.650
32 is 2 times 16.

00:01:04.650 --> 00:01:06.660
Once again, 16's a perfect
square, so

00:01:06.660 --> 00:01:08.490
we could stop there.

00:01:08.490 --> 00:01:10.160
If you didn't realize that,
you would factor

00:01:10.160 --> 00:01:11.230
that as 4 times 4.

00:01:11.230 --> 00:01:12.260
You'd see that twice.

00:01:12.260 --> 00:01:14.980
You could even go even further
down to 2 times 2 and all of

00:01:14.980 --> 00:01:16.940
that, but you see immediately
that's a perfect square, so we

00:01:16.940 --> 00:01:18.020
can stop there.

00:01:18.020 --> 00:01:22.030
So this second expression can
be written as minus 6 times

00:01:22.030 --> 00:01:28.920
the square root of 16 times
the square root of 2.

00:01:28.920 --> 00:01:31.510
This right here-- I want to be
clear-- is the same thing as

00:01:31.510 --> 00:01:34.520
the square root of 16 times 2.

00:01:34.520 --> 00:01:35.770
You can separate out.

00:01:35.770 --> 00:01:38.850
The square root of 16 times 2 is
the square root of 16 times

00:01:38.850 --> 00:01:40.000
the square root of 2.

00:01:40.000 --> 00:01:42.600
We saw that with our exponent
properties.

00:01:42.600 --> 00:01:45.000
Now, what does this first
term simplify to?

00:01:45.000 --> 00:01:46.270
This is 3 clearly.

00:01:46.270 --> 00:01:48.230
This right here is a 2.

00:01:48.230 --> 00:01:51.050
So you have 3 times 2 times
the square root of 2.

00:01:51.050 --> 00:01:55.060
That is 6 times the principal
root of 2.

00:01:55.060 --> 00:01:57.880
And then from that we're going
to subtract-- well, what's

00:01:57.880 --> 00:01:58.790
this term right here?

00:01:58.790 --> 00:02:01.260
That is positive 4.

00:02:01.260 --> 00:02:06.820
So 6 times 4 is 24 times
the square root of 2.

00:02:06.820 --> 00:02:08.240
And we're not done yet.

00:02:08.240 --> 00:02:11.780
If I have 6 of something and
I'm going to subtract from

00:02:11.780 --> 00:02:14.910
that 24 of that same something,
what do I have?

00:02:14.910 --> 00:02:17.290
I have 6 square roots of 2 and
I'm going to subtract from

00:02:17.290 --> 00:02:20.780
that 24 square roots of 2,
well, this is going to be

00:02:20.780 --> 00:02:28.020
equal to 6 minus 24 is negative
18 square roots of 2.

00:02:28.020 --> 00:02:29.420
And hopefully, this doesn't
confuse you.

00:02:29.420 --> 00:02:35.250
Remember, if we had 6x minus
24x, we would have minus 18x

00:02:35.250 --> 00:02:37.150
or negative 18x.

00:02:37.150 --> 00:02:38.870
Now, instead of an x, we just
have a square root of 2.

00:02:38.870 --> 00:02:42.090
6 of something minus 24 of
something will get us negative

00:02:42.090 --> 00:02:44.120
18 of that something.

00:02:44.120 --> 00:02:45.880
Let's do another one.

00:02:45.880 --> 00:02:53.300
Let's say I have the square root
of 180 plus 6 times the

00:02:53.300 --> 00:02:56.470
square root of 405.

00:02:56.470 --> 00:02:59.940
So this is really an exercise
in being able to simplify

00:02:59.940 --> 00:03:01.600
these radicals, which
we've done before.

00:03:01.600 --> 00:03:04.250
But you can never get too much
practice doing that.

00:03:04.250 --> 00:03:06.230
So let's just do the
factorization

00:03:06.230 --> 00:03:07.610
of these right here.

00:03:07.610 --> 00:03:14.670
So 180 is 2 times 90,
which is 2 times 45,

00:03:14.670 --> 00:03:18.290
which is 5 times 9.

00:03:18.290 --> 00:03:21.860
And we can factor 9 down more
into 3 times 3 to realize it's

00:03:21.860 --> 00:03:23.550
a perfect square, but we could
leave it like that.

00:03:23.550 --> 00:03:27.940
So this first term right here
we can write as the square

00:03:27.940 --> 00:03:34.550
root of 2 times 2 times the
square root of 5 times the

00:03:34.550 --> 00:03:37.420
square root of 9.

00:03:37.420 --> 00:03:39.120
I'm going to put the square
root of 9 out front.

00:03:39.120 --> 00:03:41.470
So square root of 2 times 2
times the square root of 5

00:03:41.470 --> 00:03:45.440
times the square root of 9.

00:03:45.440 --> 00:03:48.360
Now, what is this second
term equal to?

00:03:48.360 --> 00:03:49.900
So let's factor it out.

00:03:49.900 --> 00:03:50.870
405.

00:03:50.870 --> 00:03:54.670
That is 5 times--
I think it's 81.

00:03:54.670 --> 00:04:00.970
But just to verify, 405,
5 doesn't go into 4, so

00:04:00.970 --> 00:04:02.340
let's go into 40.

00:04:02.340 --> 00:04:04.040
5 goes into 40 eight times.

00:04:04.040 --> 00:04:06.340
8 times 5 is 40.

00:04:06.340 --> 00:04:07.090
Subtract.

00:04:07.090 --> 00:04:07.910
You get a 0.

00:04:07.910 --> 00:04:09.520
Bring down the 5.

00:04:09.520 --> 00:04:11.470
5 goes into 5 one time.

00:04:11.470 --> 00:04:14.100
Right, 81 times.

00:04:14.100 --> 00:04:17.290
81 is 9 times 9.

00:04:17.290 --> 00:04:20.260
You could factor more if we were
trying to do the fourth

00:04:20.260 --> 00:04:22.390
root or something like that,
but we want to just do a

00:04:22.390 --> 00:04:22.960
square root.

00:04:22.960 --> 00:04:25.910
We have a 9 and a 9, so no
need to factor any more.

00:04:25.910 --> 00:04:31.360
So this second expression right
here is plus 6 times the

00:04:31.360 --> 00:04:40.600
square root of 9 times 9 times
the square root of 5.

00:04:40.600 --> 00:04:41.260
So what is this?

00:04:41.260 --> 00:04:43.140
This is 3.

00:04:43.140 --> 00:04:44.710
This is 2.

00:04:44.710 --> 00:04:45.860
This is the square root of 4.

00:04:45.860 --> 00:04:48.270
So it's 3 times 2 is 6.

00:04:48.270 --> 00:04:51.790
So we have 6 square
roots of 5.

00:04:51.790 --> 00:04:54.160
Plus-- what's this right here?

00:04:54.160 --> 00:04:57.100
The square root of 9 times
9, the square root of 81.

00:04:57.100 --> 00:04:59.380
That's, of course, just 9.

00:04:59.380 --> 00:05:09.190
So 6 times 9 is 54, so plus
54 square roots of 5.

00:05:09.190 --> 00:05:12.520
And then, what do
we have left?

00:05:12.520 --> 00:05:17.120
We have 6 of something
plus 54 of something.

00:05:17.120 --> 00:05:22.050
That's going to be equal
to 60 of that

00:05:22.050 --> 00:05:24.400
something just like that.

00:05:24.400 --> 00:05:27.490
Let's just do one more and
we're going to have some

00:05:27.490 --> 00:05:28.750
abstract quantities here.

00:05:28.750 --> 00:05:30.000
We're going to deal with
some variables.

00:05:30.000 --> 00:05:31.960
But I really just want to do
it to show you that the

00:05:31.960 --> 00:05:34.200
variables don't change
anything.

00:05:34.200 --> 00:05:36.520
Let's say if we have the square
root or the principal

00:05:36.520 --> 00:05:38.300
root of 48a.

00:05:38.300 --> 00:05:46.830
And I'm going to add that to
the square root of 27a.

00:05:46.830 --> 00:05:50.130
So once again, let's just
factor the 48 part.

00:05:50.130 --> 00:05:51.930
We'll leave the a aside.

00:05:51.930 --> 00:05:57.350
So 48 is 2 times 24, which
is 2 times 12.

00:05:57.350 --> 00:06:04.650
Sorry, 2 times 12, which
is 3 times 4.

00:06:04.650 --> 00:06:08.250
So we could rewrite this first
expression here as the square

00:06:08.250 --> 00:06:14.920
root of 2 times 2 times the
square root of 4 times the

00:06:14.920 --> 00:06:16.590
square root of 3.

00:06:16.590 --> 00:06:18.520
Now, you might have done
it a quicker way.

00:06:18.520 --> 00:06:21.040
You might have just factored
into 3 and 16 and immediately

00:06:21.040 --> 00:06:23.090
realized that 16 is
a perfect square.

00:06:23.090 --> 00:06:25.190
But I did it just kind of
the brute force way.

00:06:25.190 --> 00:06:27.470
You'd get the same answer
either way.

00:06:27.470 --> 00:06:29.950
And, of course, not just the
square root of 3, you also

00:06:29.950 --> 00:06:31.210
have the square root
of a there.

00:06:31.210 --> 00:06:32.910
So I'll just put the
a right over here.

00:06:32.910 --> 00:06:35.060
I could put it in a separate
square root, but both of these

00:06:35.060 --> 00:06:37.710
aren't perfect squares, so I'll
leave both of these under

00:06:37.710 --> 00:06:39.250
the radical sign.

00:06:39.250 --> 00:06:43.750
Now, 27 is 3 times 9.

00:06:43.750 --> 00:06:46.040
9 is a perfect square root,
so we can stop there.

00:06:46.040 --> 00:06:49.480
So this second term, we can
rewrite it as the square root

00:06:49.480 --> 00:06:54.295
of 9 times the square
root of 3a.

00:06:54.295 --> 00:06:56.620
And in both of these you can
kind of view it I'm skipping

00:06:56.620 --> 00:06:57.750
an intermediate step.

00:06:57.750 --> 00:07:01.970
The intermediate step, I could
have written that first

00:07:01.970 --> 00:07:08.070
expression as the square root
of 9 times 3a and then

00:07:08.070 --> 00:07:09.130
gone to this step.

00:07:09.130 --> 00:07:12.210
But I think we have enough
practice realizing that 9

00:07:12.210 --> 00:07:16.510
times 3a, all of that to the
1/2 power, or taking the

00:07:16.510 --> 00:07:18.970
principal root of all of that
is the same thing as taking

00:07:18.970 --> 00:07:23.130
the principal root of 9 times
the principal root of 3a.

00:07:23.130 --> 00:07:25.080
So that's the step I skipped
in both of these.

00:07:25.080 --> 00:07:27.580
But hopefully, that doesn't
confuse you too much.

00:07:27.580 --> 00:07:30.140
And so, this term right here
is going to be a 2.

00:07:30.140 --> 00:07:31.990
This term right here
is going to be a 2.

00:07:31.990 --> 00:07:37.220
So this is going to be 4 times
the square root of 3a.

00:07:37.220 --> 00:07:40.840
And then this over here, this
right here, is a 3.

00:07:40.840 --> 00:07:45.000
So this is going to be plus 3
times the square root of 3a.

00:07:45.000 --> 00:07:51.480
4 of something plus 3 of
something will be equal to 7

00:07:51.480 --> 00:07:53.930
of the something.

00:07:53.930 --> 00:07:56.270
Anyway, hopefully, you
found that useful.


Graphs of square-root functions

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.730
I think you're probably
reasonably familiar with the

00:00:02.730 --> 00:00:05.790
idea of a square root, but I
want to clarify some of the

00:00:05.790 --> 00:00:08.250
notation that at least, I always
found a little bit

00:00:08.250 --> 00:00:10.000
ambiguous at first. I
want to make it very

00:00:10.000 --> 00:00:13.100
clear in your head.

00:00:13.100 --> 00:00:18.680
If I write a 9 under a radical
sign, I think you know you'll

00:00:18.680 --> 00:00:20.940
read this as the square
root of 9.

00:00:20.940 --> 00:00:22.880
But I want to make one
clarification.

00:00:22.880 --> 00:00:25.990
When you just see a number
under a radical sign like

00:00:25.990 --> 00:00:38.650
this, this means the principal
square root of 9.

00:00:38.650 --> 00:00:41.420
And when I say the principal
square root, I'm really saying

00:00:41.420 --> 00:00:44.300
the positive square root of 9.

00:00:44.300 --> 00:00:49.770
So this statement right
here is equal to 3.

00:00:49.770 --> 00:00:52.350
And I'm being clear here because
you might already know

00:00:52.350 --> 00:00:57.390
that 9 has two actual
square roots.

00:00:57.390 --> 00:01:01.000
By definition, a square root is
something-- A square root

00:01:01.000 --> 00:01:04.190
of 9 is a number that, if
you square it, equals 9.

00:01:04.190 --> 00:01:07.830
3 is a square root, but
so is negative 3.

00:01:07.830 --> 00:01:14.260
Negative 3 is also
a square root.

00:01:14.260 --> 00:01:17.310
But if you just write a radical
sign, you're actually

00:01:17.310 --> 00:01:19.990
referring to the positive square
root, or the principal

00:01:19.990 --> 00:01:20.650
square root.

00:01:20.650 --> 00:01:22.710
If you want to refer to the
negative square root, you'd

00:01:22.710 --> 00:01:25.520
actually put a negative in front
of the radical sign.

00:01:25.520 --> 00:01:28.170
That is equal to negative 3.

00:01:28.170 --> 00:01:30.970
Or if you wanted to refer to
both the positive and the

00:01:30.970 --> 00:01:34.520
negative, both the principal and
the negative square roots,

00:01:34.520 --> 00:01:38.450
you'll write a plus or
a minus sign in front

00:01:38.450 --> 00:01:39.220
of the radical sign.

00:01:39.220 --> 00:01:44.650
And of course, that's equal to
plus or minus 3 right there.

00:01:44.650 --> 00:01:47.450
So with that out of the way,
what I want to talk about is

00:01:47.450 --> 00:01:54.280
the graph of the function, y
is equal to the principal

00:01:54.280 --> 00:01:56.520
square root of x.

00:01:56.520 --> 00:02:02.765
And see how it relates to the
function y is equal to x-- Let

00:02:02.765 --> 00:02:04.740
me write it over here because
I'll work on it.

00:02:04.740 --> 00:02:09.960
See how it relates to y
is equal to x squared.

00:02:09.960 --> 00:02:12.090
And then, if we have some time,
we'll shift them around

00:02:12.090 --> 00:02:15.940
a little bit and get a better
understanding of what causes

00:02:15.940 --> 00:02:19.400
these functions to shift up
down or left and right.

00:02:19.400 --> 00:02:22.590
So let's make a little value
table before we get out our

00:02:22.590 --> 00:02:24.490
graphing calculator.

00:02:24.490 --> 00:02:27.180
So this is for y is equal
to x squared.

00:02:27.180 --> 00:02:29.720
So we have x and y values.

00:02:29.720 --> 00:02:31.870
This is y is equal to the
square root of x.

00:02:31.870 --> 00:02:36.780
Once again, we have x and
y values right there.

00:02:36.780 --> 00:02:40.790
So let me just pick some
arbitrary x values right here,

00:02:40.790 --> 00:02:43.380
and I'll stay in the
positive x domain.

00:02:43.380 --> 00:02:48.300
So let's say x is equal
to 0, 1-- Let me

00:02:48.300 --> 00:02:48.900
make it color coded.

00:02:48.900 --> 00:02:51.590
When x is equal to 0, what's
y going to be equal to?

00:02:51.590 --> 00:02:53.090
Well y is x squared.

00:02:53.090 --> 00:02:55.250
0 squared is 0.

00:02:55.250 --> 00:03:00.750
When x is 1, y is 1 squared,
which is 1.

00:03:00.750 --> 00:03:06.710
When x is 2, y is 2 squared,
which is 4.

00:03:06.710 --> 00:03:12.510
When x is 3, y is 3 squared,
which is 9.

00:03:12.510 --> 00:03:13.240
We've seen this before.

00:03:13.240 --> 00:03:15.100
And I could keep going.

00:03:15.100 --> 00:03:16.340
Let me add 4 here.

00:03:16.340 --> 00:03:19.640
So when x is 4, y is
4 squared, or 16.

00:03:19.640 --> 00:03:20.430
We've seen all of this.

00:03:20.430 --> 00:03:22.000
We've graphed our parabolas.

00:03:22.000 --> 00:03:23.680
This is all a bit of review.

00:03:23.680 --> 00:03:26.800
Now let's see what happens
when y is equal to the

00:03:26.800 --> 00:03:29.300
principal square root of x.

00:03:29.300 --> 00:03:30.650
Let's see what happens.

00:03:30.650 --> 00:03:33.360
And I'm going to pick some x
values on purpose just to make

00:03:33.360 --> 00:03:34.200
it interesting.

00:03:34.200 --> 00:03:37.790
When x is equal to 0, what's
y going to be equal to?

00:03:37.790 --> 00:03:39.220
The principal square
root of 0?

00:03:39.220 --> 00:03:40.550
Well it's 0.

00:03:40.550 --> 00:03:42.410
0 squared is 0.

00:03:42.410 --> 00:03:46.280
When x is equal to 1, the
principal square root of x of

00:03:46.280 --> 00:03:48.110
1 is just positive 1.

00:03:48.110 --> 00:03:50.050
It has another square root
that's negative 1, but we

00:03:50.050 --> 00:03:53.000
don't have a positive or
negative written here.

00:03:53.000 --> 00:03:55.000
We just have the principal
square root.

00:03:55.000 --> 00:03:59.500
When x is a 4, what is y?

00:03:59.500 --> 00:04:03.340
Well, the principal square
root of 4 is positive 2.

00:04:03.340 --> 00:04:06.430
When x is equal to
9, what's y?

00:04:06.430 --> 00:04:08.630
When x is equal to 9,
the principal square

00:04:08.630 --> 00:04:09.980
root of 9 is 3.

00:04:09.980 --> 00:04:12.950
Finally, when x is equal to 16,
the principal square root

00:04:12.950 --> 00:04:14.960
of 16 is 4.

00:04:14.960 --> 00:04:17.720
So I think you already see how
these two are related.

00:04:17.720 --> 00:04:21.920
We've essentially just swapped
the x's and the y's.

00:04:21.920 --> 00:04:24.250
Well, these are the same x and
y's, but here you have

00:04:24.250 --> 00:04:26.510
x is 2, y is 4.

00:04:26.510 --> 00:04:29.020
Here x is 4, y is 2.

00:04:29.020 --> 00:04:31.480
3 comma 9, 9 comma 3.

00:04:31.480 --> 00:04:34.730
4 comma 16, 16 comma 4.

00:04:34.730 --> 00:04:35.960
And that makes complete sense.

00:04:35.960 --> 00:04:42.430
If you were to square both sides
of this equation, you

00:04:42.430 --> 00:04:48.150
would get y squared is equal
to x right there.

00:04:48.150 --> 00:04:51.530
And, of course, you would want
to restrict the domain of y to

00:04:51.530 --> 00:04:55.480
positive y's because this can
only take on positive values

00:04:55.480 --> 00:04:57.020
because this is a principal
square root.

00:04:57.020 --> 00:05:00.090
But the general idea, we just
swapped the x's and y's

00:05:00.090 --> 00:05:03.580
between this function and this
function right here, if you

00:05:03.580 --> 00:05:07.620
assume a domain of positive
x's and positive y's.

00:05:07.620 --> 00:05:09.550
Now, let's see what the
graphs look like.

00:05:09.550 --> 00:05:12.730
And I think you might already
have a guess of-- Let me just

00:05:12.730 --> 00:05:13.960
graph them here.

00:05:13.960 --> 00:05:16.450
Let me do them by hand because
I think that's instructive

00:05:16.450 --> 00:05:19.160
sometimes before you take out
the graphing calculator.

00:05:19.160 --> 00:05:22.180
So I'm just going to stay in
the positive, in the first

00:05:22.180 --> 00:05:25.260
quadrant here.

00:05:25.260 --> 00:05:27.800
So let me graph this first.

00:05:27.800 --> 00:05:33.740
So we have the point 0, 0, the
point 1 comma 1, the point 2

00:05:33.740 --> 00:05:36.550
comma 2, which I'm going to have
to draw it a little bit

00:05:36.550 --> 00:05:37.590
smaller than that.

00:05:37.590 --> 00:05:41.280
Let me mark this is 1, 2, 3.

00:05:41.280 --> 00:05:42.870
Actually, let me do
it like this.

00:05:42.870 --> 00:05:49.890
Let me go 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11,

00:05:49.890 --> 00:05:53.010
12, 13, 14, 15, 16.

00:05:53.010 --> 00:05:54.070
That's about how far
I have to go.

00:05:54.070 --> 00:06:02.340
And then I have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13,

00:06:02.340 --> 00:06:04.610
14, 15, 16.

00:06:04.610 --> 00:06:07.000
That's about how far I have to
go in that direction as well.

00:06:07.000 --> 00:06:08.060
And now let's graph it.

00:06:08.060 --> 00:06:14.580
So we have 0, 0, 1, 1,
2 comma 1, 2, 3, 4.

00:06:14.580 --> 00:06:17.590
2 comma 4 right there.

00:06:17.590 --> 00:06:18.720
3 comma 9.

00:06:18.720 --> 00:06:22.410
3 comma 5, 6, 7, 8, 9.

00:06:22.410 --> 00:06:26.060
3 comma 9 is right
about there.

00:06:26.060 --> 00:06:28.450
And then we have 4 comma 16.

00:06:28.450 --> 00:06:32.660
4 comma 16 is going to
be right above there.

00:06:32.660 --> 00:06:34.770
So the graph of y is equal
to x squared, and

00:06:34.770 --> 00:06:36.200
we've seen this before.

00:06:36.200 --> 00:06:39.180
It's going to look something
like this.

00:06:39.180 --> 00:06:41.740
We're just graphing it in the
positive quadrant, so we get

00:06:41.740 --> 00:06:44.010
this upward opening
u just like that.

00:06:47.100 --> 00:06:49.870
Now let's graph y is equal
to the principal

00:06:49.870 --> 00:06:52.210
square root of x.

00:06:52.210 --> 00:06:54.750
So here, once again,
we have 0, 0.

00:06:54.750 --> 00:06:56.490
We have 1 comma 1.

00:06:56.490 --> 00:06:57.900
We have 4 comma 2.

00:06:57.900 --> 00:07:01.580
1, 2, 3, 4 comma 2.

00:07:01.580 --> 00:07:03.140
We have 9 comma 3.

00:07:03.140 --> 00:07:08.750
5, 6, 7, 8, 9 comma 3
right about there.

00:07:08.750 --> 00:07:10.690
Then we have 16 comma 4.

00:07:10.690 --> 00:07:13.390
16 comma 4 is right
about there.

00:07:13.390 --> 00:07:18.620
So this graph looks like that.

00:07:18.620 --> 00:07:21.480
So notice, they look like
they're kind of

00:07:21.480 --> 00:07:23.230
flipped around the axes.

00:07:23.230 --> 00:07:26.370
This one opens along the y-axis,
this one opens along

00:07:26.370 --> 00:07:27.230
the x-axis.

00:07:27.230 --> 00:07:30.110
And once again, it makes
complete sense because we've

00:07:30.110 --> 00:07:31.680
swapped the x's and the y's.

00:07:31.680 --> 00:07:34.090
Especially if you just consider
the first quadrant.

00:07:34.090 --> 00:07:38.460
And actually, these are
symmetric around the line, y

00:07:38.460 --> 00:07:39.523
is equal to x.

00:07:39.523 --> 00:07:42.350
And we'll talk about things like
inverses in the future

00:07:42.350 --> 00:07:45.850
that are symmetric around the
line, y is equal to x.

00:07:45.850 --> 00:07:48.580
And we can graph this better
on a regular graphing

00:07:48.580 --> 00:07:49.720
calculator.

00:07:49.720 --> 00:07:50.960
I found this on the web.

00:07:50.960 --> 00:07:52.425
I just did a quick web search.

00:07:52.425 --> 00:07:55.800
I want to give proper credit
to the people whose

00:07:55.800 --> 00:07:57.360
resource I'm using.

00:07:57.360 --> 00:08:02.370
So this is my.hrw.com/math06.

00:08:02.370 --> 00:08:03.350
You could pause this video.

00:08:03.350 --> 00:08:04.580
And hopefully, you should
be able read this.

00:08:04.580 --> 00:08:06.490
Especially if you're looking
at it in HD.

00:08:06.490 --> 00:08:08.850
But let's graph these
different things.

00:08:08.850 --> 00:08:10.270
Let's graph it a little
bit cleaner than

00:08:10.270 --> 00:08:11.600
what I can do by hand.

00:08:11.600 --> 00:08:15.030
And actually, let me have some
of what I wrote there.

00:08:15.030 --> 00:08:15.360
So that

00:08:15.360 --> 00:08:18.590
should give you-- OK.

00:08:18.590 --> 00:08:24.390
So let's first just graph
y is equal to x squared.

00:08:24.390 --> 00:08:27.770
And then in green, let me
graph y is equal to the

00:08:27.770 --> 00:08:30.720
square root of x.

00:08:30.720 --> 00:08:33.070
They have some buttons here on
the right, just so you know

00:08:33.070 --> 00:08:33.650
what I'm doing.

00:08:33.650 --> 00:08:36.000
I have some buttons here on
the right: squared and the

00:08:36.000 --> 00:08:39.030
radical sign and all of that.

00:08:39.030 --> 00:08:41.110
Let me just focus on this.

00:08:41.110 --> 00:08:43.900
So let me just graph those.

00:08:43.900 --> 00:08:46.250
So first it did x squared
and then it did the

00:08:46.250 --> 00:08:47.050
square root of x.

00:08:47.050 --> 00:08:50.500
Look, if you just focus on the
first quadrant right here, you

00:08:50.500 --> 00:08:53.680
see that you get the exact same
result that I got over

00:08:53.680 --> 00:08:56.120
there, although mine
is messier.

00:08:56.120 --> 00:08:59.150
Now, just for fun and, you know,
I really didn't do this

00:08:59.150 --> 00:09:00.590
yet with the regular
quadratics,

00:09:00.590 --> 00:09:01.730
let's see what happens.

00:09:01.730 --> 00:09:04.770
What we need to do to shift
the different graphs.

00:09:04.770 --> 00:09:07.250
So with x squared, I'm going
to do two things.

00:09:07.250 --> 00:09:09.990
I'm going to scale the graphs
and I'm going to shift them.

00:09:09.990 --> 00:09:10.800
So that's x squared.

00:09:10.800 --> 00:09:13.190
So let's just focus on the x
squared and see what happens

00:09:13.190 --> 00:09:14.030
when we scale it.

00:09:14.030 --> 00:09:16.610
And then I'll do it with the
radical sign as well.

00:09:16.610 --> 00:09:17.930
This will really work
for anything.

00:09:17.930 --> 00:09:24.480
Let's see what happens when you
get 2 times-- no, not 2

00:09:24.480 --> 00:09:28.050
squared --2 times x squared.

00:09:28.050 --> 00:09:36.070
And let's do another one that is
1.5 times our 0.-- I could

00:09:36.070 --> 00:09:37.470
just do 0.4 actually.

00:09:37.470 --> 00:09:41.290
0.5 times x squared.

00:09:41.290 --> 00:09:44.350
Let's graph these right there.

00:09:44.350 --> 00:09:45.190
So x squared.

00:09:45.190 --> 00:09:48.290
So notice, our regular x
squared is just in red.

00:09:48.290 --> 00:09:51.610
If we scale it by 2, it's still
a parabola with the

00:09:51.610 --> 00:09:54.970
vertex at the same place, but
we go up faster in both

00:09:54.970 --> 00:09:55.790
directions.

00:09:55.790 --> 00:09:58.780
And if we have 0.5 times x
squared, we still have a

00:09:58.780 --> 00:10:01.680
parabola, but we go up
a little bit slower.

00:10:01.680 --> 00:10:06.230
We have a wider opening u
because our scaling factor is

00:10:06.230 --> 00:10:07.570
lower than 1.

00:10:07.570 --> 00:10:11.530
So that's how you kind of decide
how wide or how narrow

00:10:11.530 --> 00:10:13.660
the opening of our
parabola is.

00:10:13.660 --> 00:10:15.990
And then if you want to shift
it to the left or the right,

00:10:15.990 --> 00:10:18.410
and I want you to think
about why this is.

00:10:18.410 --> 00:10:20.030
So that's x squared.

00:10:20.030 --> 00:10:21.690
Let's say I want to just take
the graph of x squared and I

00:10:21.690 --> 00:10:25.040
want to shift it four
to the right.

00:10:25.040 --> 00:10:28.396
What I do is I say, x minus 4.

00:10:28.396 --> 00:10:30.820
x minus 4 squared.

00:10:30.820 --> 00:10:33.640
And if I want to shift it two
to the-- Let's say I want to

00:10:33.640 --> 00:10:36.980
shift it two to the
left. x plus 2

00:10:36.980 --> 00:10:41.350
squared, what do we get?

00:10:41.350 --> 00:10:44.570
Notice it did exactly what I
said. x minus 4 squared was

00:10:44.570 --> 00:10:45.960
shifted four to the right.

00:10:45.960 --> 00:10:48.750
x plus 2 squared, was shifted
two to the left.

00:10:48.750 --> 00:10:51.740
And it might be unintuitive at
first, this shifting that I'm

00:10:51.740 --> 00:10:52.530
talking about.

00:10:52.530 --> 00:10:54.420
But really think about
what's happening.

00:10:54.420 --> 00:10:57.320
Over here, the vertex is
where x is equal to 0.

00:10:57.320 --> 00:11:00.320
When you get 0 squared
up here.

00:11:00.320 --> 00:11:02.810
Now over here, the vertex
is when x is equal to 4.

00:11:02.810 --> 00:11:05.430
But when x is equal to 4,
you stick 4 in here,

00:11:05.430 --> 00:11:07.260
you get 4 minus 4.

00:11:07.260 --> 00:11:10.230
So you're still squaring 0.

00:11:10.230 --> 00:11:13.410
4 minus 4 is 0 and that's
what you're squaring.

00:11:13.410 --> 00:11:18.050
Over here, when x is equal to
negative 2-- negative 2 plus 2

00:11:18.050 --> 00:11:20.360
--you are squaring 0.

00:11:20.360 --> 00:11:24.370
So, in other words, whatever
you're squaring, that 0 is

00:11:24.370 --> 00:11:25.900
equivalent to 4 here.

00:11:25.900 --> 00:11:27.660
Or 4 is equivalent to 0.

00:11:27.660 --> 00:11:29.790
And negative 2 is equivalent
to 0 over there.

00:11:29.790 --> 00:11:30.990
So I want you to think about
it a little bit.

00:11:30.990 --> 00:11:33.000
Another way you could think
about it, when x is equal to

00:11:33.000 --> 00:11:35.490
1, we're at this point
of the red parabola.

00:11:35.490 --> 00:11:39.070
But when x is equal to 5 on
the green parabola, you

00:11:39.070 --> 00:11:41.010
have 5 minus 4.

00:11:41.010 --> 00:11:45.060
Inside of the parentheses you
have a 1, just like x is equal

00:11:45.060 --> 00:11:47.250
to 1 over here, up here.

00:11:47.250 --> 00:11:49.550
So you're at the same point
in the parabola.

00:11:49.550 --> 00:11:50.910
So I want you to think about
that a little bit.

00:11:50.910 --> 00:11:52.840
It might be a little
non-intuitive that you say

00:11:52.840 --> 00:11:55.560
minus 4 to shift to the
right and plus 2 to

00:11:55.560 --> 00:11:56.380
shift to the left.

00:11:56.380 --> 00:11:58.650
But it actually makes
a lot of sense.

00:11:58.650 --> 00:12:00.820
Now, the other interesting
thing is to shift

00:12:00.820 --> 00:12:01.950
things up and down.

00:12:01.950 --> 00:12:03.760
And that's actually pretty
straightforward.

00:12:03.760 --> 00:12:06.200
You want to shift
this curve up.

00:12:06.200 --> 00:12:08.400
Let's say we want to shift the
red curve up a little bit.

00:12:08.400 --> 00:12:10.930
You do x squared plus 1.

00:12:10.930 --> 00:12:12.130
Notice it got shifted up.

00:12:12.130 --> 00:12:16.090
If you want this green curve to
be shifted down by 5, put a

00:12:16.090 --> 00:12:17.850
minus 5 right there.

00:12:17.850 --> 00:12:21.350
And then you graph it and it
got shifted down by 5.

00:12:21.350 --> 00:12:25.620
If you want it to open up a
little wider than that, maybe

00:12:25.620 --> 00:12:27.320
scale it down a little bit.

00:12:27.320 --> 00:12:30.110
Scale it down and let's
say 0.5 times that.

00:12:30.110 --> 00:12:34.440
So now the green curve will be
scaled down and it opens

00:12:34.440 --> 00:12:36.570
slower, it has a
wider opening.

00:12:36.570 --> 00:12:39.510
And the same idea can
be done with the

00:12:39.510 --> 00:12:41.020
principal square roots.

00:12:41.020 --> 00:12:43.510
So let me do that.

00:12:43.510 --> 00:12:45.180
Let me do the same idea.

00:12:45.180 --> 00:12:48.340
And the same idea actually, can
be done with any function.

00:12:48.340 --> 00:12:51.260
So let's do the square
root of x.

00:12:51.260 --> 00:12:55.570
And in green, let's do
the square root of x.

00:12:55.570 --> 00:12:57.140
Let's say, minus 5.

00:12:57.140 --> 00:12:59.540
So we're shifting it over
to the right by 5.

00:12:59.540 --> 00:13:04.355
And then let's have the square
root of x plus 4.

00:13:04.355 --> 00:13:07.650
So we're going to shift
it to the left by 4.

00:13:07.650 --> 00:13:10.860
Let's shift it down by 3.

00:13:10.860 --> 00:13:13.060
And so lets graph
all of these.

00:13:13.060 --> 00:13:14.110
The square root of x.

00:13:14.110 --> 00:13:15.730
Then have the square
root of x minus 5.

00:13:15.730 --> 00:13:18.470
Notice it's the exact same thing
as the square root of x,

00:13:18.470 --> 00:13:21.760
but I shifted it to
the right by 5.

00:13:21.760 --> 00:13:24.730
When x is equal to 5, I have
a 0 under the radical sign.

00:13:24.730 --> 00:13:26.400
Same thing as square
root of 0.

00:13:26.400 --> 00:13:29.070
So this point is equivalent
to that point.

00:13:29.070 --> 00:13:33.030
Now, when I have the square root
of x plus 4, I've shifted

00:13:33.030 --> 00:13:34.630
it over to the left by 4.

00:13:34.630 --> 00:13:38.000
When x is negative 4, I have
a 0 under the radical sign.

00:13:38.000 --> 00:13:40.030
So this point is equivalent
to that point.

00:13:40.030 --> 00:13:43.140
And then I subtracted 3, which
also shifted it down 3.

00:13:43.140 --> 00:13:44.810
So this is my starting point.

00:13:44.810 --> 00:13:51.660
If I want this blue square root
to open up slower, so

00:13:51.660 --> 00:13:53.070
it'll be a little
bit narrower, I

00:13:53.070 --> 00:13:54.740
would scale it down.

00:13:54.740 --> 00:13:57.900
So here, putting a low number
will scale it down and make it

00:13:57.900 --> 00:14:01.770
more narrow because we're
opening along the x-axis.

00:14:01.770 --> 00:14:04.430
So let me to do that.

00:14:04.430 --> 00:14:07.250
Let me make this green one--
Let me open up wider.

00:14:07.250 --> 00:14:11.460
So let me say it's 3 times the
square root of x minus 5.

00:14:11.460 --> 00:14:13.780
So let's graph all of these.

00:14:13.780 --> 00:14:16.660
So notice, this blue one now
opens up more narrow and this

00:14:16.660 --> 00:14:19.640
green one now opens up
a lot, I guess you

00:14:19.640 --> 00:14:20.470
could say, a lot faster.

00:14:20.470 --> 00:14:21.480
It's scaled up.

00:14:21.480 --> 00:14:25.410
Then we could shift that one
up a little bit by 4.

00:14:25.410 --> 00:14:28.000
And then we graph it
and there you go.

00:14:28.000 --> 00:14:31.240
And notice when we graph these,
it's not a sideways

00:14:31.240 --> 00:14:32.120
parabola because we're
talking about the

00:14:32.120 --> 00:14:33.540
principal square root.

00:14:33.540 --> 00:14:35.450
And if you did the plus or minus
square root, it actually

00:14:35.450 --> 00:14:37.020
wouldn't even be a valid
function because you would

00:14:37.020 --> 00:14:39.720
have two y values for
every x value.

00:14:39.720 --> 00:14:42.780
So that's why we have to just
use the principal square root.

00:14:42.780 --> 00:14:45.940
Anyway, hopefully you found
this little talk, I guess,

00:14:45.940 --> 00:14:50.900
about the relationships with
parabolas, and/or with the x

00:14:50.900 --> 00:14:53.550
squared's and the principal
square roots, useful.

00:14:53.550 --> 00:14:54.340
And how to shift them.

00:14:54.340 --> 00:14:56.210
And that will actually be really
useful in the future

00:14:56.210 --> 00:14:59.840
when we talk about inverses
and shifting functions.


Radical expressions with higher roots

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.590
So far, when we were dealing
with radicals we've only been

00:00:03.590 --> 00:00:05.370
using the square root.

00:00:05.370 --> 00:00:09.190
We've seen that if I write a
radical sign like this and put

00:00:09.190 --> 00:00:13.180
a 9 under it, this means the
principal square root of 9,

00:00:13.180 --> 00:00:14.690
which is positive 3.

00:00:14.690 --> 00:00:17.880
Or you could view it as the
positive square root of 9.

00:00:17.880 --> 00:00:20.840
Now, what's implicit when we
write it like this is that I'm

00:00:20.840 --> 00:00:23.360
taking the square root.

00:00:23.360 --> 00:00:25.670
So I could have also written
it like this.

00:00:25.670 --> 00:00:27.640
I could have also written the
radical sign like this and

00:00:27.640 --> 00:00:32.040
written this index 2 here, which
means the square root,

00:00:32.040 --> 00:00:35.570
the principal square
root of 9.

00:00:35.570 --> 00:00:37.680
Find me something that
if I square that

00:00:37.680 --> 00:00:40.130
something, I get 9.

00:00:40.130 --> 00:00:42.330
And the radical sign doesn't
just have to

00:00:42.330 --> 00:00:44.750
apply to a square root.

00:00:44.750 --> 00:00:48.210
You can change the index here
and then take an arbitrary

00:00:48.210 --> 00:00:49.660
root of a number.

00:00:49.660 --> 00:00:54.390
So for example, if I were to
ask you, what-- You could

00:00:54.390 --> 00:00:57.040
imagine this is called the cube
root, or you could call

00:00:57.040 --> 00:01:00.780
it the third root of 27.

00:01:00.780 --> 00:01:01.980
What is this?

00:01:01.980 --> 00:01:05.120
Well, this is some number that
if I take it to the third

00:01:05.120 --> 00:01:07.910
power, I'd get 27.

00:01:07.910 --> 00:01:09.420
Well, the only number that if
you take it to the third

00:01:09.420 --> 00:01:12.540
power, you get 27
is equal to 3.

00:01:12.540 --> 00:01:18.420
3 times 3 times 3
is equal to 27.

00:01:18.420 --> 00:01:23.040
9 times 3, 27.

00:01:23.040 --> 00:01:25.720
So likewise, let me
just do one more.

00:01:25.720 --> 00:01:29.760
So if I have 16-- I'll do
it in a different color.

00:01:29.760 --> 00:01:34.780
If I have 16 and I want to take
the fourth root of 16,

00:01:34.780 --> 00:01:38.840
what number times itself
4 times is equal to 16?

00:01:38.840 --> 00:01:41.030
And if it doesn't pop out at
you immediately, you can

00:01:41.030 --> 00:01:43.920
actually just do a prime
factorization of 16

00:01:43.920 --> 00:01:45.250
to figure it out.

00:01:45.250 --> 00:01:45.440
Let's see.

00:01:45.440 --> 00:01:47.830
16 is 2 times 8.

00:01:47.830 --> 00:01:50.120
8 is 2 times 4.

00:01:50.120 --> 00:01:52.340
4 is 2 times 2.

00:01:52.340 --> 00:01:58.010
So this is equal to the fourth
root of 2 times 2

00:01:58.010 --> 00:01:59.910
times 2 times 2.

00:01:59.910 --> 00:02:01.510
You have these four 2's here.

00:02:01.510 --> 00:02:05.450
Well, I have four 2's being
multiplied, so the fourth root

00:02:05.450 --> 00:02:10.039
of this must be equal to 2.

00:02:10.039 --> 00:02:11.750
And you could also view this
as kind of the fourth

00:02:11.750 --> 00:02:15.750
principal root because if these
were all negative 2's,

00:02:15.750 --> 00:02:17.100
it would also work.

00:02:21.890 --> 00:02:24.360
Just like you have multiple
square roots, you have

00:02:24.360 --> 00:02:25.540
multiple fourth roots.

00:02:25.540 --> 00:02:29.020
But the radical sign implies
the principal root.

00:02:29.020 --> 00:02:32.530
Now, with that said, we've
simplified traditional square

00:02:32.530 --> 00:02:33.510
roots before.

00:02:33.510 --> 00:02:36.940
Now we should hopefully be able
to simplify radicals with

00:02:36.940 --> 00:02:38.155
higher power roots.

00:02:38.155 --> 00:02:39.980
So let's try a couple.

00:02:39.980 --> 00:02:42.790
Let's say I want to simplify
this expression.

00:02:42.790 --> 00:02:47.140
The fifth root of 96.

00:02:47.140 --> 00:02:50.860
So like I said before, let's
just factor this right here.

00:02:50.860 --> 00:02:55.080
So 96 is 2 times 48.

00:02:55.080 --> 00:02:58.000
Which is 2 times 24.

00:02:58.000 --> 00:03:00.640
Which is 2 times 12.

00:03:00.640 --> 00:03:03.160
Which is 2 times 6.

00:03:03.160 --> 00:03:05.970
Which is 2 times 3.

00:03:05.970 --> 00:03:12.790
So this is equal to the fifth
root of 2 times 2 times 2

00:03:12.790 --> 00:03:14.040
times 2 times 2.

00:03:20.090 --> 00:03:23.310
Times 3.

00:03:23.310 --> 00:03:26.520
Or another way you could view
it, is you could view it to a

00:03:26.520 --> 00:03:28.540
fractional power.

00:03:28.540 --> 00:03:30.540
You could view it to
a fractional power.

00:03:30.540 --> 00:03:31.720
We've talked about
that already.

00:03:31.720 --> 00:03:36.980
This is the same thing as 2
times 2 times 2 times 2 times

00:03:36.980 --> 00:03:41.780
2 times 3 to the 1/5 power.

00:03:41.780 --> 00:03:44.330
Let me make this clear.

00:03:44.330 --> 00:03:49.270
Having an nth root of some
number is equivalent to taking

00:03:49.270 --> 00:03:52.560
that number to the 1/n power.

00:03:52.560 --> 00:03:55.510
These are equivalent statements
right here.

00:03:55.510 --> 00:03:57.640
So if you're taking this to
the 1/5 power, this is the

00:03:57.640 --> 00:04:01.220
same thing as taking 2 times
2 times 2 times 2

00:04:01.220 --> 00:04:03.640
times 2 to the 1/5.

00:04:03.640 --> 00:04:06.150
Times 3 to the 1/5.

00:04:06.150 --> 00:04:08.760
Now I have something
that's multiplied.

00:04:08.760 --> 00:04:11.630
I have 2 multiplied
by itself 5 times.

00:04:11.630 --> 00:04:13.280
And I'm taking that
to the 1/5.

00:04:13.280 --> 00:04:15.180
Well, the 1/5 power of this
is going to be 2.

00:04:15.180 --> 00:04:17.570
Or the fifth root of this
is just going to be 2.

00:04:17.570 --> 00:04:19.220
So this is going to
be a 2 right here.

00:04:19.220 --> 00:04:21.899
And this is going to be
3 to the 1/5 power.

00:04:21.899 --> 00:04:26.860
2 times 3 to the 1/5, which is
this simplified about as much

00:04:26.860 --> 00:04:27.750
as you can simplify it.

00:04:27.750 --> 00:04:29.790
But if we want to keep in
radical form, we could write

00:04:29.790 --> 00:04:39.020
it as 2 times the fifth
root 3 just like that.

00:04:39.020 --> 00:04:40.270
Let's try another one.

00:04:44.320 --> 00:04:46.480
Let me put some variables
in there.

00:04:46.480 --> 00:04:53.850
Let's say we wanted to simplify
the sixth root of 64

00:04:53.850 --> 00:04:56.570
times x to the eighth.

00:04:56.570 --> 00:04:59.420
So let's do 64 first.

00:04:59.420 --> 00:05:06.230
64 is equal to 2 times 32,
which is 2 times 16.

00:05:06.230 --> 00:05:07.980
Which is 2 times 8.

00:05:07.980 --> 00:05:09.610
Which is 2 times 4.

00:05:09.610 --> 00:05:11.860
Which is 2 times 2.

00:05:11.860 --> 00:05:14.730
So we have 1, 2, 3, 4, 5, 6.

00:05:14.730 --> 00:05:16.900
So it's essentially 2
to the sixth power.

00:05:16.900 --> 00:05:21.880
So this is equivalent to the
sixth root of 2 to the sixth--

00:05:21.880 --> 00:05:28.530
that's what 64 is --times
x to the eighth power.

00:05:28.530 --> 00:05:31.500
Now, the sixth root of 2 to
the sixth, that's pretty

00:05:31.500 --> 00:05:33.140
straightforward.

00:05:33.140 --> 00:05:39.290
So this part right here is just
going to be equal to 2.

00:05:39.290 --> 00:05:47.460
That's going to be 2 times
the sixth root of x

00:05:47.460 --> 00:05:48.710
to the eighth power.

00:05:51.230 --> 00:05:53.300
And how can we simplify this?

00:05:53.300 --> 00:05:58.310
Well, x to the eighth power,
that's the same thing as x to

00:05:58.310 --> 00:06:02.860
the sixth power times
x squared.

00:06:02.860 --> 00:06:04.630
You have the same base, you
would add the exponents.

00:06:04.630 --> 00:06:07.050
This is the same thing
as x to the eighth.

00:06:07.050 --> 00:06:11.960
So this is going to be equal to
2 times the sixth root of x

00:06:11.960 --> 00:06:14.940
to the sixth times x squared.

00:06:14.940 --> 00:06:18.500
And the sixth root, this part
right here, the sixth root of

00:06:18.500 --> 00:06:20.770
x to the sixth, that's just x.

00:06:20.770 --> 00:06:26.270
So this is going to be equal to
2 times x times the sixth

00:06:26.270 --> 00:06:30.900
root of x squared.

00:06:30.900 --> 00:06:33.810
Now, we can simplify this
even more if you

00:06:33.810 --> 00:06:34.570
really think about.

00:06:34.570 --> 00:06:37.990
Remember, this expression right
here, this is the exact

00:06:37.990 --> 00:06:44.130
same thing as x squared
to the 1/6 power.

00:06:44.130 --> 00:06:47.100
And if you remember your
exponent properties, when you

00:06:47.100 --> 00:06:49.620
raise something to an exponent,
and then raise that

00:06:49.620 --> 00:06:53.800
to an exponent, that's
equivalent to x to

00:06:53.800 --> 00:06:56.490
the 2 times 1/6 power.

00:06:56.490 --> 00:07:00.720
Or-- let me write this --2 times
1/6 power, which is the

00:07:00.720 --> 00:07:05.010
same thing-- Let me not forget
to write my 2x there.

00:07:05.010 --> 00:07:08.080
So I have a 2x there
and a 2x there.

00:07:08.080 --> 00:07:12.300
And this is the same thing as
2x-- it's the same 2x there

00:07:12.300 --> 00:07:16.810
--times x to the 2/6.

00:07:16.810 --> 00:07:20.220
Or, if we want to write that in
most simple form or lowest

00:07:20.220 --> 00:07:26.710
common form, you get 2x times
x to the-- What do you have

00:07:26.710 --> 00:07:28.870
here? x to the 1/3.

00:07:28.870 --> 00:07:31.320
So if you want to write it in
radical form, you could write

00:07:31.320 --> 00:07:38.180
this is equal to 2 times 2x
times the third root of x.

00:07:38.180 --> 00:07:42.720
Or, the other way to think about
it, you could just say--

00:07:42.720 --> 00:07:46.650
So we could just go from
this point right here.

00:07:46.650 --> 00:07:47.580
We could write this.

00:07:47.580 --> 00:07:49.190
We could ignore this,
what we did before.

00:07:49.190 --> 00:07:53.120
And we could say, this is the
same thing as 2 times x to the

00:07:53.120 --> 00:07:56.090
eighth to the 1/6 power.

00:07:56.090 --> 00:07:58.650
x to the eighth to
the 1/6 power.

00:07:58.650 --> 00:08:03.570
So this is equal to 2
times x to the-- 8

00:08:03.570 --> 00:08:06.710
times 1/6 --8/6 power.

00:08:06.710 --> 00:08:09.360
Now we can reduce
that fraction.

00:08:09.360 --> 00:08:13.780
That's going to be 2 times
x to the 4/3 power.

00:08:13.780 --> 00:08:17.610
And this and this are completely
equivalent.

00:08:17.610 --> 00:08:18.660
Why is that?

00:08:18.660 --> 00:08:22.530
Because we have 2 times x or 2
times x to the first power

00:08:22.530 --> 00:08:24.320
times x to the 1/3 power.

00:08:24.320 --> 00:08:28.470
You add 1 to 1/3, you get 4/3.

00:08:28.470 --> 00:08:31.350
So hopefully you found this
little tutorial on higher

00:08:31.350 --> 00:08:32.585
power radicals interesting.

00:08:32.585 --> 00:08:35.570
And I think it is useful to kind
of see it in prime factor

00:08:35.570 --> 00:08:37.640
form and realize, oh, if I'm
taking the sixth root, I have

00:08:37.640 --> 00:08:40.330
to find a prime factor that
shows up at least six times.

00:08:40.330 --> 00:08:42.210
And then I could figure out
that's 2 to the sixth.

00:08:42.210 --> 00:08:45.600
Anyway, hopefully you found
this mildly useful.