We were asked to simplify
the given radical expressions.
A radical expression is not considered simplified
if there's a fraction under the radical
or if there's a radical in the denominator.
For the first example we have the square root of 100X
divided by five X.
The first step should always be to see
if the fraction under the radical will simplify.
100X divided by five X simplifies very nicely
because five X is a factor of 100X.
To show this, let's write 100X
as 20 times five X.
Notice in this form, we can see that
five X divided by five X will simplify to one
and therefore the radical simplifies
to the square root of 20,
but now we need to simplify the square root of 20
by identifying any perfect square factors of 20.
To do this, let's look at the prime factorization of 20,
which is two times two times five.
So the square root of 20 is equal to the square root
of two times two times five.
Notice here we have two factors of two,
which shows 20 contains the perfect square factor
of two squared or four.
let's write two times two as two squared.
This is equal to the square root of two squared times five
and the square root of two squared simplifies perfectly
to one factor of two.
The simplified expression is two square root of five.
Next we have the cube root
of 64X squared Y to the fifth
divided by the cube root of four Y squared.
Now this radical expression looks rather messy.
So what we'll do in this case is use the radical property
shown below in the opposite direction.
Notice the property indicates the X root of A
divided by the X root of B
is equal to the X root of the fraction A over B,
which means the quotient of these two cube roots
is equal to the cube root
of the fraction 64X squared Y to the fifth
divided by four Y squared.
In this form, we will now simplify the fraction
and then simplify the resulting cube root.
So we have the cube root of 64 divided by four
is equal to 16.
Then we have X squared.
Then we have Y to the fifth divided by Y to the second.
Remember, when dividing and the bases are the same,
we subtract the exponent.
So Y to the fifth divided by Y to the second
is equal to Y raised to the power of five minus two,
which equals Y cubed.
So this simplifies here
the cube root of 16X squared Y to the third.
Now to continue simplifying, because we have a cube root,
we look for perfect cube factors of the radicand.
Well the prime factorization of 16
is four factors of two.
So we have two times two times two times two.
Notice how here we have three equal factors of two
and we have X squared,
which does not contain any perfect square factors.
Let's go ahead and write X squared as X times X.
Then we have Y to the third,
which is equal to Y times Y times Y.
Notice here we have three equal factors of Y.
Let's write this one more time using exponents
before we simplify.
Let's write this as the cube root
of two cubed times two times the X squared,
which will not simplify, times Y cubed.
And again, here we have three equal factors of two
and three equal factors of Y.
The cube root of two cubed is equal to one factor of two.
The cube root of Y cubed is equal to one factor of Y
and we still have the cube root of two X squared.
This is the simplified radical expression
and because the index is odd,
there's no need for an absolute value.
I hope you found this helpful.