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>> Welcome to the Cypress
College Math Review on Radicals.
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Objective one, taking nth roots with numbers.
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B is the nth root of a if b to the nth power
is equal to a. Here's a notation that we use
00:00:21.526 --> 00:00:31.766
for radicals b is the nth root of a, n is called
the index (the little number up to the left
00:00:31.906 --> 00:00:36.856
of the radical sign), a is the
radicand (the thing inside the radical)
00:00:37.556 --> 00:00:41.616
and the whole thing nth root of a
is called the radical expression.
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Here's an example, 2 is equal to
the cube root or 3rd root of 8.
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Two is the 3rd root or cube root of 8.
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Three is the index of the radical.
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Eight is the radicand -- the
thing inside the radical.
00:01:04.206 --> 00:01:08.796
And the cube root of 8 is the
radical, the whole radical expression.
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In order to work with radicals, it's helpful to
know as many perfect square numbers as possible.
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Try to memorize as many of
these relationships as you can.
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Now, if b to the nth is equal to a then
b is equal to the nth root of a. Well,
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let's put some numbers in here
to make that a little more clear.
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If 3 squared is equal to 9, which it is, 3
times 3 is nine, then 3 is a 2nd root of 9.
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Well, how do we write 2nd root?
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Second root is the same thing as square root.
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In other words, when the index of the
radical is a 2, we don't write it.
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So, 3 is the square root of 9.
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So, this is a 3.
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So, what would you square to get 4?
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Two. What would you square to get 1?
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One. What would you square to get 16?
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Four. What would you square to get 25?
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Five. These are the square
roots of those numbers.
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Now, let's look at cube roots.
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Once again, it's helpful to know as
many perfect cube numbers as possible.
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So, learn as many of these
relationships as you can.
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What would you cube to get 1?
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Well, that's just 1.
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So, the cube root of 1 is just 1.
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What would you cube to get 8?
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Well, that's 2, so the cube root of 8 is 2.
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What would you cube to get 27?
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Well, that would be 3, 3 cubed is 27.
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So, the cube root of 27 is 3.
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What would you cube to get 64?
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Well, that's 4, so the cube root of 64 is 4.
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What would you cube to get 125?
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That's 5, so the cube root of 125 is 5.
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Example, simplify the following radical.
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The square root of 49.
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Well, the square root of 49 is 7.
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Why? Because 7 squared is equal to 49.
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Maybe you didn't know that.
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Maybe you didn't know that perfect square.
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What could you do then?
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You could do a factor tree.
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So, you take the number 49 and you do a factor
tree and it turns out that it's 7 times 7.
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So, you write the square root of
49 as the square root of 7 times 7.
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Now, to be a perfect square, a number
must have two of the same factor.
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We have two of the same factor.
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When this happens, you can
take those two factors together
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and allow one to escape the square root.
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So, the square root of 7 times 7 is just 7.
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Now, note that you do not write
the radical symbol anymore.
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Now we want to take the 3rd
root or cube root of 216.
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Well, that's equal to 6.
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Why? Because if you take 6
and cube it, you get 216.
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We could rewrite 216 as 6 times 6 times 6.
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To be a perfect cube a number must
have three of the same factor.
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When this happens, we can take
those three factors together
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and allow one to escape the cube root.
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So, three factors together to escape
a cube root so we get the number 6.
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Principal and negative square roots.
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If you take the number 5 and you square it.
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You get 25.
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If you take the number negative 5
and square that you also get 25.
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Well that means there's actually
two different square roots
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of the number 25 or any positive number.
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So, 5 is a square root of 25 and
negative 5 is a square root of 25.
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What does this symbol stand for then?
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This symbol stands for just the
principal square root of 25.
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The non-negative square root.
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So, the symbol stands for just the
non-negative or principal square root of 25.
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If you put a negative in front of that symbol,
that gives you the other square root of 25.
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The square root of a negative
number is not a real number.
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So, the square root of negative 9 is not real.
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The square root of 0 is 0 because
if you square 0 you get 0.
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The square root of 1 is 1
because 1 squared is 1.
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If the index is even such as, square root
which is the same thing as a 2nd root,
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4th root or 6th root, then the radicand must be
non-negative for the root to be a real number.
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For example, the 4th root of 16 is equal
to 2, because 2 to the 4th is equal to 16.
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But the 4th root of negative
16 is not a real number.
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A lot of people would say oh negative 2 works.
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But no, if you take negative 2 and raise it
to the 4th power, you have an even number
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of negatives, which is going to give
you positive 16, not negative 16.
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So, the 4th root of negative
16 is not a real number.
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The 6th root of 64 is equal to 2
because, 2 to the 6th is equal to 64.
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But the 6th root of negative
64 is not a real number.
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If the index is odd, such as cube root,
which is 3rd root, 5th root or 7th root,
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then the radicand can be any real number.
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For example, the cube root of 64 is equal
to 4 because 4 cubed is equal to 64.
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The cube root of negative 64 is
negative because if you take negative 4
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and cube it that's an odd number of
negatives, so that does give you a negative 64.
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The 5th root of 32 is equal to 2
because 2 to the 5th is equal to 32.
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And the 5th root of negative 32 is equal to
negative 2 because if you take negative 2
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and raise it to an odd power
you will get a negative number.
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If we do not get a perfect power that
matches the index of the radical,
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we can still take out any
double factors that exist
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and whatever's leftover will
be stuck under the radical.
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So, we're working with 18.
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Eighteen is 2 times 9 and 9 is 3 times 3.
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So, we do have a double factor in the
3's so, we have 2 times 3 times 3.
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So, the pair of 3's will come out and give us
one 3 on the outside and the 2 is left inside.
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Forty is 4 times 10.
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Four is 2 times 2 and 10 is 2 times 5.
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We're working with the cube root of 40.
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So, we have the cube root of
2 times 2 times 2 times 5.
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We need three like factors to be
able to come out of a cube root.
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So, the cube root of 2 cubed is 2 it comes
outside and we're left with a 5 on the inside.
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Be sure to write the index
of the radical, the little 3.
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So, this is 2 times the cube root of 5.
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Well, 162 is 2 times 81.
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Eighty-one is 9 times 9 and 9 is 3 times 3.
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So, here we have the square root of
2 times 3 times 3 times 3 times 3.
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So, we have pair of 3's, so one 3 will escape.
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Then we have another pair of 3's
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so another 3 will escape.
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The 2 is left inside.
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So, the answer is 9 times the square root of 2.
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Pause the video and try these problems.
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Objective two, taking nth roots with variables.
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Variables follow the same rules that numbers do.
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It takes two like factors to escape
the square root, three like factors
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to escape the cube root and so on and so forth.
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So, let's take the square root of x squared.
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Well, the square root of x
squared is equal to x. Why?
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Because if you square x you get x squared.
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What's the square root of x to the 4th?
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Well that's x squared, why?
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Because when you square x
squared you get x to the 4th.
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The square root of x to the 6th is x cubed
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because if you square x cubed
you get x to the 6th.
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So, what's happening here?
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If you take the exponent and
divide by the index of the radical
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and for a square root the
index of the radical is 2.
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You get the new exponent.
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So, 8 divided by 2 is 4.
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So, the square root of x
to the 8th is x to the 4th.
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Ten divided by 2 is 5.
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So, the square root of x to
the 10th is x to the 5th.
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Let's take cube roots of variables.
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You can rewrite this as x times itself three
times and since we're doing a cube root,
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it takes three like factors to pull it outside.
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So, that gives us just x to the 1st on
the outside and nothing left inside.
00:12:19.236 --> 00:12:23.256
You can also think about this in
terms of dividing the exponents.
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If you take the exponent that's inside divide
by the index of the radical and we get 1.
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And that's the exponent you're left with.
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Let's take the cube root of x to the 6th.
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So, 6 divided by 3 is 2.
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Because not too many people want
to keep writing out all these x's.
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So, there's one x that would come out.
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There's another x that would come out.
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Multiply times each other, that
gives us x to the 2nd power.
00:12:58.800 --> 00:13:05.550
Nine divided by 3 is 3, so the cube
root of x to the 9th is x to the 3rd.
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Twelve divided by 3 is 4.
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Fifteen divided by 3 is 5.
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Simplify the following radicals, assume
variables represent positive real numbers.
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Since this is a square root
the index of the radical is 2,
00:13:28.036 --> 00:13:30.726
so we divide by 4 by 2 and we get 2.
00:13:31.316 --> 00:13:35.286
On this problem, we divide 10 by 2 and we get 5.
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Now, we're going to divide 11 by
2, we get 5 with a remainder of 1.
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So, you will end up with 5 x's on the
outside and one that's left on the inside.
00:13:53.736 --> 00:13:58.086
The remainder tells you the power
of the variable left inside.
00:13:58.806 --> 00:14:02.086
The quotient tells you how
many x's are on the outside.
00:14:03.236 --> 00:14:06.976
The other way to do this problem is
to write out all of those 11 x's.
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Then, two like factors will
bring us an x on the outside,
00:14:21.226 --> 00:14:30.626
another pair of like factors gives us another
x, and another, and another, and another.
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Those are all on the outside of the radical.
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Left inside is that one x there.
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Once again, that gives us the answer we have
above x to the 5th times the square root
00:14:46.596 --> 00:14:51.546
of x. Cube root of x to the 17th.
00:14:52.386 --> 00:14:58.396
So, we divide our exponent 17 by
the index of the radical which is 3.
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We get 5 with the remainder of 2.
00:15:05.056 --> 00:15:08.696
So, we will end up with x to
the 5th power on the outside
00:15:09.176 --> 00:15:11.616
and x to the 2nd power on the inside.
00:15:13.536 --> 00:15:18.596
In general, for any division problem the
remainder is always smaller than the divisor.
00:15:18.756 --> 00:15:21.176
In this case, 2 of course is smaller than 3.
00:15:22.636 --> 00:15:25.196
For a radical to be in simplified form,
00:15:25.196 --> 00:15:30.966
the exponents on the inside must be
smaller than index of the radical.
00:15:31.366 --> 00:15:33.646
Once again, 2 of course is smaller than 3.
00:15:34.426 --> 00:15:42.736
For this problem, we divide 23 by
the index of the radical which is 5
00:15:43.366 --> 00:15:46.676
and we get 4 with the remainder of 3.
00:15:47.880 --> 00:15:56.500
So, the answer would be x to the 4th
times the 5th root of x to the 3rd.
00:15:58.376 --> 00:16:01.366
This problem has numbers and variables.
00:16:01.866 --> 00:16:07.466
We've done the prime factorization of the
coefficient that was there and we've written it
00:16:07.466 --> 00:16:09.996
as 3 to the 5th times 5 to the 6th.
00:16:10.436 --> 00:16:15.106
So, we can work with this just like
the 3 and the 5 were variables.
00:16:16.166 --> 00:16:24.076
So, we divide 5 by 4 and
get 1 with a remainder of 1.
00:16:25.326 --> 00:16:33.286
We divide 6 by 4 and we get
1 with a remainder of 2.
00:16:34.616 --> 00:16:44.296
We divide 11 by 4 and get
2 with a remainder of 3.
00:16:44.816 --> 00:16:55.556
We divide 12 by 4 and get 3 with no
remainder, so no y's are left on the inside.
00:16:55.696 --> 00:17:04.396
So, our answer is 15 x squared, y cubed
times the 4th root of - now what is this,
00:17:04.396 --> 00:17:11.986
5 squared is 25 and 25 times
3 is 75, so 75 x cubed.
00:17:12.976 --> 00:17:18.786
So, if you have your coefficient written
in terms of a prime factorization,
00:17:18.786 --> 00:17:24.406
you have it all factored, you can work
with it just like you can variables.
00:17:25.016 --> 00:17:32.156
We have 96 and we want to break
it down into its prime factors.
00:17:32.626 --> 00:17:38.216
So, that 3 times 32 and 32 is made up all 2's.
00:17:38.546 --> 00:17:39.926
We've seen this number before.
00:17:40.546 --> 00:17:51.826
So, we have the 5th root of 2
to the 5th power times 3 times a
00:17:51.826 --> 00:17:56.116
to the 45th, times b to the 37th.
00:17:57.416 --> 00:17:59.826
Now we divide the exponents by the index.
00:17:59.826 --> 00:18:06.036
Five divided by 5 is 1 and there's no
remainder, so there's no 2's left on the inside.
00:18:07.286 --> 00:18:10.976
Three is just to the exponent
of 1, which is already smaller
00:18:10.976 --> 00:18:13.616
than the index, so the 3 stays on the inside.
00:18:14.916 --> 00:18:21.576
Forty-five divided by 5 is 9, with no
remainder, so there's no a's left on the inside.
00:18:22.900 --> 00:18:30.720
Thirty-seven divided by 5 is 7, with a
remainder of 2 and that's our answer.
00:18:33.440 --> 00:18:36.080
Pause the video and try these problems.
00:18:40.040 --> 00:18:42.760
Objective three, adding and
subtracting radicals.
00:18:43.796 --> 00:18:47.926
Like radicals have the same
index and the same radicand.
00:18:48.946 --> 00:18:53.666
We add or subtract radicals similar
to the way that we combine like terms.
00:18:54.436 --> 00:18:57.786
Add or subtract the coefficients
then keep the radical.
00:18:58.666 --> 00:19:02.226
Three x plus 7x is equal to 10x.
00:19:03.176 --> 00:19:06.936
Three times the square root of 5 plus
7 times the square root of 5 is equal
00:19:06.936 --> 00:19:09.156
to 10 times the square root of 5.
00:19:09.666 --> 00:19:15.466
Three of these plus 7 of these is 10 of these.
00:19:16.766 --> 00:19:18.476
If the radicals are not the same,
00:19:18.796 --> 00:19:22.116
then we simplify them first
before we combine like terms.
00:19:22.656 --> 00:19:27.396
On this problem, the radicals are the
same, they're both the square root of 7.
00:19:27.806 --> 00:19:32.496
Four of these minus 9 of
these is negative 5 of these.
00:19:33.406 --> 00:19:37.976
On the next problem, the radicals are
also the same, the cube root of 2y.
00:19:38.866 --> 00:19:45.276
Nine of these plus 15 of these is 24, of these.
00:19:49.226 --> 00:19:53.416
These radicals are not the same
so first we have to simplify them.
00:19:54.376 --> 00:20:02.156
The prime factorization of 50 is 5
square times 2, because 25 times 2.
00:20:03.116 --> 00:20:08.946
The prime factorization of 18
is 3 square times 2, 9 times 2.
00:20:09.746 --> 00:20:14.146
Now we divide the exponents, these are
square roots that we're going to divide by 2.
00:20:15.156 --> 00:20:20.476
Two divided by 2 is 1, with no remainder,
so no 5's are left on the inside.
00:20:21.316 --> 00:20:22.976
The exponent for 2 is a 1,
00:20:23.206 --> 00:20:27.106
which is smaller than the index
of the radical, which is 2.
00:20:27.886 --> 00:20:32.956
Therefore, the square root of 2 stays on
the inside, because 1 is smaller than 2.
00:20:34.416 --> 00:20:41.386
We divide 2 by 2 and we get 3 to the
1st on the outside with no remainder.
00:20:42.076 --> 00:20:43.986
The 2 is left on the inside.
00:20:45.246 --> 00:20:50.606
So, we have 5 times the square root
of 2 plus 15 times the square root 2.
00:20:51.046 --> 00:20:58.426
Well 5 of these plus 15 of these is 20 of these.
00:20:59.026 --> 00:21:02.006
For this problem, we're not
going to do prime factorizations.
00:21:02.776 --> 00:21:07.886
Here we recognize that 4 is a perfect
square and it is a factor of 20.
00:21:08.646 --> 00:21:11.556
That leaves us 5x on the inside.
00:21:13.066 --> 00:21:17.446
Sixteen is a perfect square,
leaves x on the inside.
00:21:19.626 --> 00:21:24.076
Nine is a perfect square and that
is a factor of 45 which leaves us
00:21:24.076 --> 00:21:29.366
with 5 times x. The square root of 4 is 2.
00:21:29.896 --> 00:21:34.106
The square root of 16 is 4.
00:21:35.736 --> 00:21:37.546
The square root of 9 is 3.
00:21:38.146 --> 00:21:45.936
So, we have 2 times the square root
of 5x minus 24 times the square root
00:21:45.936 --> 00:21:49.976
of x plus 3 times the square root of 5x.
00:21:50.556 --> 00:21:52.176
Which of these are like terms?
00:21:52.766 --> 00:21:54.316
Well the first and the last.
00:21:55.086 --> 00:22:00.076
So, we have 5 times the square
root of 5x the other term is not a
00:22:00.076 --> 00:22:02.486
like term and it cannot be combined.
00:22:03.156 --> 00:22:10.236
Here we recognize that the cube root of 8 is 2.
00:22:10.236 --> 00:22:11.306
So, we bring that out.
00:22:11.746 --> 00:22:14.866
You could also write that out as
2 cubed and then once again you end
00:22:14.866 --> 00:22:16.426
up with 2 to the 1st on the outside.
00:22:17.696 --> 00:22:25.366
Divide the exponent 5 by 3 and
we get 1 with a remainder of 2.
00:22:25.996 --> 00:22:31.776
The cube root of 27 is 3 because 27 is 3 cubed.
00:22:33.516 --> 00:22:39.876
We divide 5 by 3 and we get
1 with a remainder of 2.
00:22:40.266 --> 00:22:41.716
These are like terms.
00:22:42.396 --> 00:22:46.396
Not only does the radical have to be
the same but since this has a variable
00:22:46.396 --> 00:22:48.716
in front, that has to be the same also.
00:22:48.716 --> 00:22:50.706
This whole expression has to be the same.
00:22:51.300 --> 00:22:59.806
We have two of these plus three of
these which gives us five of these.
00:23:01.980 --> 00:23:03.860
Pause the video and try these problems.
00:23:07.300 --> 00:23:10.840
Objective four, rationalizing
monomial denominators.
00:23:11.556 --> 00:23:15.186
First let's look at the rules for
a radical to be in simplified form.
00:23:15.616 --> 00:23:17.116
So, we're taking the nth root.
00:23:18.676 --> 00:23:22.426
First you can't have any perfect
nth powers in the radicand.
00:23:23.386 --> 00:23:32.576
So, if we had 2 to the 4th times x
squared, we would have to rewrite that as 2
00:23:32.576 --> 00:23:38.176
to the 1st times the cube
root of 2 times x squared.
00:23:38.826 --> 00:23:41.986
Notice that the exponents
that are left on the inside
00:23:41.986 --> 00:23:45.956
of the radical are smaller
than the index of the radical.
00:23:47.366 --> 00:23:52.986
That's making sure that we don't have
any perfect cubes inside a cube root.
00:23:54.486 --> 00:23:58.276
Next, you can't have any
fractions in the radicand.
00:23:58.966 --> 00:24:01.506
So, this is not simplified form.
00:24:02.776 --> 00:24:05.376
You can't leave any radicals in the denominator.
00:24:06.136 --> 00:24:08.436
This is also not simplified form.
00:24:09.376 --> 00:24:12.786
We're going to work on these two processes next.
00:24:13.546 --> 00:24:20.916
We'll use the product rule and quotient
rule for radicals to do these problems.
00:24:21.036 --> 00:24:25.476
Rationalizing the denominator of a
fraction with one term in the denominator.
00:24:26.386 --> 00:24:28.216
The process of eliminating a radical
00:24:28.216 --> 00:24:31.216
in the denominator is called
rationalizing the denominator.
00:24:31.906 --> 00:24:37.366
If the denominator is a single term, we
multiply both the numerator and the denominator
00:24:37.366 --> 00:24:43.296
so that the denominator is a perfect square,
or a perfect cube, or perfect 4th root,
00:24:43.336 --> 00:24:45.076
depending upon the index of the radical.
00:24:46.336 --> 00:24:50.986
Examples, simplify, since each of
these expressions have a radical
00:24:50.986 --> 00:24:55.066
in the denominator simplify includes
rationalizing the denominator.
00:24:55.866 --> 00:24:58.426
Assume variables represent
positive real numbers.
00:24:59.046 --> 00:25:01.976
On the first problem, we have the
square root of 5 in the denominator.
00:25:02.786 --> 00:25:08.646
If we multiply both the top and
the bottom by the square root of 5,
00:25:08.866 --> 00:25:14.396
we end up with by the product rule, the
square root of 5 squared in the denominator.
00:25:14.946 --> 00:25:19.806
Well, the square root of 5 squared is simply 5.
00:25:20.906 --> 00:25:24.876
So, the 5 is no longer in the radical and
we don't have a radical in the denominator.
00:25:25.636 --> 00:25:30.646
We check to see if 2/5 reduces
and it doesn't, so we're done.
00:25:31.296 --> 00:25:34.776
For the next problem, we
have the cube root of 1/5.
00:25:34.846 --> 00:25:40.756
First of all, since we have a radical of a
fraction, we see if the fraction can be reduced.
00:25:41.216 --> 00:25:43.436
It can't, 1/5 cannot be reduced.
00:25:43.676 --> 00:25:46.606
So, we split it using the quotient rule.
00:25:46.956 --> 00:25:50.256
The cube root of 1 over the cube root of 5.
00:25:51.356 --> 00:25:53.006
The cube root of 1 is just one.
00:25:54.916 --> 00:25:58.036
Now we have a radical in the
denominator and we can't leave that.
00:25:58.626 --> 00:26:02.296
We have 5 to the 1st power under a cube root.
00:26:03.266 --> 00:26:09.846
We will need two more 5's, 5 to
the 2nd power so that we have the 3
00:26:10.176 --> 00:26:12.536
that we need to get out of the cube root.
00:26:13.516 --> 00:26:18.286
So, the numerator is the cube root of
5 squared and 5 squared is simply 25.
00:26:18.836 --> 00:26:25.456
The denominator is the cube root of 5 cubed.
00:26:26.146 --> 00:26:29.706
What's the cube root of 5 cubed?
00:26:29.946 --> 00:26:30.876
That's just 5.
00:26:32.956 --> 00:26:38.646
And that gives us our answer, no
more radical in the denominator.
00:26:39.226 --> 00:26:44.216
For the next problem, we're doing a 4th root.
00:26:44.826 --> 00:26:52.276
So, we have x to the 1st power, how many more
x's do we need so we have x to the 4th power?
00:26:52.806 --> 00:26:54.436
Well, that would be x cubed.
00:26:54.496 --> 00:27:01.326
So, we have 6 times the 4th
root of x cubed in the numerator
00:27:01.826 --> 00:27:05.296
and the denominator we have
the 4th root of x to the 4th.
00:27:06.506 --> 00:27:16.086
But the 4th root of x to the 4th is simply
x. For the next problem, we have the 7th root
00:27:16.086 --> 00:27:18.136
of x to the 5th in the denominator.
00:27:18.776 --> 00:27:29.626
We need two more x's to make it
so that we will have a 7th root
00:27:30.046 --> 00:27:32.646
of x to the 7th in the denominator.
00:27:36.856 --> 00:27:51.006
Now our denominator is simply x. No
more radical and the problem's done.
00:27:51.366 --> 00:27:55.906
You could combine this into one
radical and you would do this
00:27:55.906 --> 00:27:58.716
if anything canceled out and simplified.
00:27:59.396 --> 00:28:01.596
But nothing does, so there's
no reason to do it that way.
00:28:02.206 --> 00:28:06.046
Let's start working on that denominator.
00:28:06.566 --> 00:28:10.876
Well, 32 is 2 to the 5th power.
00:28:13.916 --> 00:28:19.846
Now, what do we need to multiply by to make
this radical in the denominator disappear?
00:28:20.446 --> 00:28:26.636
Well, the 5th root of 2 to the 5th is just 2,
so there aren't going to be any 2's left inside.
00:28:26.636 --> 00:28:27.986
So, we don't need any more 2's.
00:28:29.486 --> 00:28:31.586
The power for b is 17.
00:28:32.036 --> 00:28:34.276
The new power needs to be a multiple of 5.
00:28:34.896 --> 00:28:37.406
Five, 10, 15, 20.
00:28:37.406 --> 00:28:40.976
Ah... 20, so I need 20 b's altogether.
00:28:40.976 --> 00:28:42.086
So, I need three more.
00:28:42.716 --> 00:28:47.436
We multiply the numerators times each other.
00:28:49.486 --> 00:28:56.556
In the denominator we have the 5th root
of 2 to the 5th times b to the 20th.
00:28:57.666 --> 00:29:00.016
In the numerator, can that be simplified?
00:29:01.106 --> 00:29:01.796
Always check.
00:29:02.886 --> 00:29:08.666
Two is smaller than 5, 3 is smaller than
5, so the numerator cannot be simplified.
00:29:08.666 --> 00:29:14.926
How about the denominator?
00:29:15.246 --> 00:29:19.516
The 5th root of 2 to the 5th is just
2, so 2 to the 1st comes outside.
00:29:20.186 --> 00:29:23.596
And 20 divided by 5 is 4.
00:29:24.166 --> 00:29:26.246
No more radical.
00:29:26.746 --> 00:29:34.696
On the next problem, you should combine them
into one radical because that fraction reduces.
00:29:35.206 --> 00:29:40.846
Divide the numerator and
denominator each by 3 and we get 6/25.
00:29:41.296 --> 00:29:44.526
Now we split it using the quotient rule.
00:29:45.196 --> 00:29:51.856
And the denominator is simply 5
because the square root of 25 is 5.
00:29:52.216 --> 00:29:53.986
No more radical in the denominator.
00:29:54.566 --> 00:29:58.226
And the square root of 6 is simplified.
00:29:58.540 --> 00:30:01.100
There's no perfect squares
that can come out of that.
00:30:03.780 --> 00:30:06.120
Pause the video and try these problems.
00:30:10.040 --> 00:30:13.620
Objective five, rationalizing
binomial denominators.
00:30:14.616 --> 00:30:20.186
Recall from factoring the formula the quantity
a plus b times the quantity a minus b is equal
00:30:20.186 --> 00:30:22.076
to a squared minus b squared.
00:30:23.276 --> 00:30:28.116
This formula plays an integral
part in allowing us to rationalize
00:30:28.426 --> 00:30:32.386
if our denominator is a binomial
involving square roots.
00:30:33.236 --> 00:30:38.116
Where a plus b and a minus b are
called conjugates of each other.
00:30:38.826 --> 00:30:41.876
First, let's multiply a couple
conjugates and see what we get.
00:30:43.046 --> 00:30:49.026
When you multiply 4 minus the square root
of 3 times 4 plus the square root of 3,
00:30:49.706 --> 00:30:56.626
you get 4 squared minus the square root
of 3 squared, a squared minus b squared.
00:30:56.626 --> 00:30:58.916
The outer and inner terms
in F.O.I.L. cancel out.
00:30:59.336 --> 00:31:04.496
So, we end up with 16 minus 3 which is 13.
00:31:04.946 --> 00:31:06.066
No more radical.
00:31:09.856 --> 00:31:14.106
Simplify. And since these expressions
have a radical in the denominator,
00:31:14.406 --> 00:31:17.646
simplify includes rationalizing the denominator.
00:31:18.696 --> 00:31:22.186
We have two terms in the
denominator and at least one
00:31:22.186 --> 00:31:25.226
of them has a square root, so
we multiply by the conjugate.
00:31:25.586 --> 00:31:28.886
The conjugate of the square
root of 5 plus square root
00:31:28.886 --> 00:31:32.136
of 3 is the square root of 5
minus the square root of 3.
00:31:32.996 --> 00:31:35.136
What you do to the top, you
must do to the bottom.
00:31:35.536 --> 00:31:38.386
Because this expression has
to equal the number 1.
00:31:38.866 --> 00:31:41.926
You can only multiply an
expression by the number 1.
00:31:42.806 --> 00:31:47.576
Since this 6 does not have a radical
in it, I leave it factored out.
00:31:48.016 --> 00:31:50.146
So, I have 6 times that quantity.
00:31:51.256 --> 00:31:54.616
In the denominator, I have
a squared minus b squared.
00:31:55.216 --> 00:32:01.456
So, I'm going to have the square root of 5
squared minus the square root of 3 squared,
00:32:01.616 --> 00:32:05.006
because the outer and the inner terms
once again in F.O.I.L., cancel out.
00:32:05.116 --> 00:32:11.016
So, we will end up with a
squared minus b squared.
00:32:11.046 --> 00:32:15.006
We leave the 6 out there just in case
there will be any canceling that we can do.
00:32:15.576 --> 00:32:18.916
So, the square root of 5 squared is just 5.
00:32:19.366 --> 00:32:22.206
The square root of 3 squared is 3.
00:32:22.206 --> 00:32:33.746
So, we end up with 2 in the denominator
and that does cancel with our 6,
00:32:34.636 --> 00:32:40.406
and since our denominator is now
just 1 we go ahead and distribute,
00:32:41.566 --> 00:32:50.596
and our final answer is 3 times the square
root of 5 minus 3 times the square root of 3.
00:32:50.846 --> 00:32:53.646
Here we also have two terms in the denominator
00:32:54.136 --> 00:32:57.736
with at least one radical, so
we multiply by the conjugate.
00:32:59.106 --> 00:33:05.916
So, 5 times the square root of 3 plus 4
times the square root of 2 over itself.
00:33:06.086 --> 00:33:08.526
Once again, you can only
multiply by the number 1.
00:33:10.556 --> 00:33:16.156
This numerator has a radical in it so, in
this case we do go ahead and distribute.
00:33:16.886 --> 00:33:19.826
So, we have 3 times the square root of 6
00:33:20.306 --> 00:33:23.846
and that's multiplied times
5 times the square root of 3.
00:33:25.256 --> 00:33:28.606
Plus 3 times the square root of 6
00:33:28.866 --> 00:33:34.236
and that's multiplied times 4 times
the square root of 2, I distributed.
00:33:35.046 --> 00:33:39.676
In the denominator, we have
a squared minus b squared.
00:33:39.706 --> 00:33:48.246
So, 5 rad 3 quantity squared
minus 4 rad 2 quantity squared.
00:33:49.946 --> 00:33:54.726
In the numerator, we have on the
outside 3 times 5 which gives us 15,
00:33:55.116 --> 00:33:58.636
on the inside we have 6 times three which is 18.
00:33:59.206 --> 00:34:04.446
For the next term in the numerator,
we have 3 times 4 which is 12.
00:34:05.136 --> 00:34:08.486
Inside the radical we have
6 times 2 which is also 12.
00:34:10.306 --> 00:34:13.816
In the denominator 5 squared is 25.
00:34:14.666 --> 00:34:17.766
The square root of 3 times
the square root of 3 is 3.
00:34:19.326 --> 00:34:22.846
Minus 4 squared is 16.
00:34:23.806 --> 00:34:26.696
The square root of 2 times
the square root of 2 is 2.
00:34:26.916 --> 00:34:28.696
No more radical in the denominator.
00:34:30.856 --> 00:34:34.306
Eighteen is 9 times 2.
00:34:35.616 --> 00:34:42.886
Twelve is 4 times 3, because we have to get
perfect squares out of those square roots.
00:34:51.096 --> 00:35:02.306
So, we have 15 times 3, square root
of 2 plus 12 times 2 square root of 3.
00:35:03.106 --> 00:35:05.506
All over 43.
00:35:06.186 --> 00:35:15.706
So, we have 45 square root of 2
plus 24 times the square root of 3.
00:35:16.356 --> 00:35:19.486
All over 43.
00:35:20.096 --> 00:35:21.806
Can we factor anything out?
00:35:21.926 --> 00:35:23.476
Yes, we can factor out a 3.
00:35:24.296 --> 00:35:25.256
What are we left with?
00:35:25.946 --> 00:35:33.766
Well, 15 times the square root of 2 plus 8
times the square root of 3 and that's over 43.
00:35:34.486 --> 00:35:37.616
Nothing cancels, so we're done.
00:35:40.096 --> 00:35:45.386
We once again have two terms in the
denominator with at least one square root,
00:35:45.386 --> 00:35:49.976
so we multiply by the conjugate
of the denominator, over itself.
00:35:55.046 --> 00:35:58.316
In our numerator, we are doing F.O.I.L. So,
00:35:58.316 --> 00:36:01.036
the first terms times each
other would be the square root
00:36:01.036 --> 00:36:04.096
of x times the square root of x or squared.
00:36:04.936 --> 00:36:07.606
The outer terms would be
10 times the square root
00:36:07.606 --> 00:36:12.556
of x. The inner terms would be
1 times the square root of x
00:36:13.226 --> 00:36:15.976
and the last terms would be 1 times 10.
00:36:17.406 --> 00:36:19.956
In the denominator, we have
a squared minus b squared.
00:36:20.676 --> 00:36:25.876
So, we have the square root
of x squared minus 10 squared.
00:36:27.800 --> 00:36:31.100
The square root of x squared those cancel
00:36:31.120 --> 00:36:38.740
and you just x. Ten rad x plus 1 rad
x is 11 rad x. We added like terms.
00:36:40.660 --> 00:36:43.580
There aren't any other like
terms in the numerator.
00:36:43.740 --> 00:36:48.000
In the denominator, we have x minus 100.
00:36:48.400 --> 00:36:54.400
Nothing factors, so we can't
cancel anything and we're done.
00:36:56.680 --> 00:36:58.920
Pause the video and try these problems.