Welcome to a lesson on Multiplying Radical
Expressions with Variables.
We saw in our previous lesson to multiply
two radical expressions when the index is the same,
in this case n, we multiply
the terms outside the radicals together, in this case
we'd have p times q.
Then we multiply the radicans together,
which are the terms under the radicals,
in this case we'd have u times v.
And then, we simplify this result if possible.
So, as an example when we have variables
if had the square root of 2x
time the square root of 3xy,
because they're both square roots
the nx's are the same, they're both 2.
In this case, the terms outside the radicals
would be ones so we just multiply the radicans.
So, we multiply 2x and 3xy under the square root
which would give us two times three
times x times x times y.
Notice in this form we can tell
we do have a perfect square factor of x squared.
So, this will simplify.
The square root of x squared would simplify
to one factor of x, so we're left with
x times the square root of six y.
Let's take a look at some more examples.
Here we have the square root of 21 x squared y
time the square root 28 xy cubed.
Again, because the coefficients would be one
we multiply the radicans, which would give us
the square root
of 21 times 28
times, we'd have x squared times x to the first.
That would be x to the third.
Times y to the first times y to the third.
That would by y to the fourth.
Now, to simplify this we do want to find
the prime factorization of 21 times 28.
So, we'll find the prime factorization
of them separately, and then combine them.
So, 21 is equal to three times seven.
28 is equal to four times seven.
And four is equal to two times two.
So, now we can write this as the square root
of, now for this prime factorization
we'll put the terms in order from least to greatest.
So, we have two times two times three
times seven times seven.
So, two times two times three
times seven times seven.
Finding the prime factorization of these two separately
saves a little bit of work than having to multiply
these together and then find the prime factorization.
And now, for x to the third we can write that
as x times x times x.
And y to the fourth would be y times y
times y times y.
Now, what we could do is recognize that x squared
is a perfect square factor and write x cubed
as x squared times x and because y squared
is also a perfect square factor we could have
written this as y squared times y squared.
But, for this example leave this expanded
and then we'll circle groups of two equal factors
since we have a square root.
Here's a perfect square factor.
Here's a perfect square factor,
as well as here, here, and here.
Everything circled will simplify.
So, we'll have the square root of 2 squared is 2.
The square root of seven squared is seven,
the square root of x squared is x.
The square root of y squared is y,
as well as the square root of this y squared.
And then, we still have the square root of three x,
So, this simplifies to
14xy squared
times the square root of three x.
Next, we have the cube root of 4 a squared b squared
times the cube root of 12 ab cubed.
Again, the coefficients are one
so we can just multiply the radicans together.
So, we'd have the cube root
of four times 12.
And then, we have a to the second times a,
that's a to the third
times b to the second times b to the third,
that would be b to the fifth.
Notice here we have a cube root,
so we're looking groups of three equal factors.
Well, the prime factorization of four
is two times two.
The prime factorization of 12 is three times four.
And, four is equal to two times two.
Again, we could have found this product is 48
and then found the prime factorization of 48,
but it's easier to find te prime factorization
in this form.
So, now we have the cube root
of, notice how we have one, two,
three, four factors of two and a factor of three.
Now, instead of expanding the variables this time
because we have a cube root we're looking for
groups of three equal factors.
So a cubed is a perfect cubed factor
so I'm going to leave that as a to the third.
And b to the third would also be a perfect cubed factor.
So, let's write b to the fifth as b to the third
times b to the second.
Now, if this is confusing we can always just expand
the variables as we did above here.
Now, we'll circle groups of three.
Here are three twos.
Here are three factors of a
and three factors of b.
So, the circled part will simplify.
This will simplify to one factor of two.
This simplifies to one factor of a.
And, this simplifies to one factor of b.
And, we're left with the cubed root
of six b squared.
So we have two ab times
the cube root of six b squared.
Let's take a look at one more example.
We have two xy square root eight x cubed y
times five square root six xy squared.
We begin by multiplying the terms
outside the square roots together.
So, 2xy times five, that's 10 x y.
And then, we have the square root
of the product of the radicans.
So we have eight times six.
And then we have x to the third
times x to the first, that's x to the fourth.
Y to the first times y to the second
that's y to the third.
So, now we want to simplify this.
So, we have 10xy times the square root of,
we know this is 48, but we want
the prime factorization.
The prime factorization of eight is two
times two times two.
The prime factorization of six
is two times three.
And now, for x to the fourth
because we're going to be circling groups
of two equal factors let's write x to the fourth
as x to the second times x to the second.
And, y to the second would also be a perfect square factor.
Let's write y to the third as y to the second
times y.
Again, if we need to we can just expand
these variable factors.
Now, we'll circle groups of two.
Here are two twos, two twos,
two factors of x, two factors of x,
two factors of y.
Those are all the perfect square factors
that will simplify.
So, we have 10xy times the square root
of two squared is two.
The square root two squared is two again.
The square root of x squared is x.
The square root of x squared is x again.
And the square root of y squared is y.
And, we're left with the square root of three y.
So, the final result would be 10 times two
times two, that's 40.
And we have x times x times x,
that's x to the third.
Y times y, that's y squared.
And, we have the square root of three y.
I hope you found this helpful.