We want to rationalize
the denominators of the given fractions.
Looking at our first example,
we have the square root of y over 45,
or y divided by 45,
which equals the square root of the numerator
over, divided by, the square root of the denominator.
In this form, because the square root of 45
does not simplify perfectly, the denominator is irrational.
To rationalize a denominator of a fraction,
we multiply the numerator and denominator of the fraction
by a radical that will clear the radical
from the denominator.
So, even though the square root of 45
does not simplify perfectly, it does simplify
because 45 contains the perfect square factor of 9.
To show this, let's look at the prime factorization of 45.
45 is equal to nine times five, five is prime,
and nine is equal to three times three,
which is the reason why nine is a
perfect square factor of 45.
So, to show some work, let's write this as
the square root of y over the square root of three
times three times five.
So, again, the square root of three times three,
or the square root of three squared,
would simplify to one factor of three
but because we have one factor of five still under
the square root, the denominator is still irrational.
In order for the denominator to simplify perfectly,
you notice how we would need two factors of five
under the square root and, therefore,
to rationalize the denominator,
we will multiply the numerator and denominator
by the square root of five.
Now, let's multiply.
The numerator would be the square root of y
times the square root of five
which is the square root of 5y.
The denominator would now be the square root of
two factors of three and two factors of five.
So, because we have two factors of three
and two factors of five, the denominator is now going to
simplify perfectly and will be rational.
So, simplifying, we have the square root of 5y over
the square root of three squared is equal to
one factor of three,
the square root of five squared is one factor of five.
So, now we have the square roof of 5y over 15
and the denominator is now rational.
Let's look at our second example.
We have the square root of two over x cubed,
or two divided by x cubed,
which is equal to the square root of two
over the square root of x cubed.
Notice how the denominator does not simplify perfectly
and, therefore, it's irrational.
Though, it does simplify because x cubed contains
a perfect square factor of x squared.
Let's go ahead and expand the factors of x cubed
and write this as the square root of two
over the square root of x times x times x
and the square root of x times x,
or the square root of x squared, does simplify
but we're still left with one factor of x
under the square root.
In order for the denominator to simplify perfectly,
you notice how we need one more factor of x
underneath the square root, and therefore,
to rationalize the denominator,
we'll multiply the numerator and denominator
by the square root of x.
Now we multiply.
In the numerator, we have the square root of two
times the square root of x,
which is the square root of 2x.
In the denominator now, we have the square root of
one, two, three, four factors of x.
So, x squared is a perfect square here, as well as here,
so now the denominator is going to simplify perfectly.
We have the square root of 2x
over the square root of x times x
to the square root of x squared here
simplifies to one factor of x.
And the same thing here.
The square root of x times x,
or the square root of x squared,
simplifies to another factor of x.
So we have the square root of 2x over,
or divided by, x squared.
And now the denominator is rationalized.
I hope you found this helpful.