We're asked to rationalize the denominator.
Here we have the square root of 2x divided by 3y.
The first step is to write the square root of the fraction
as the square root of the numerator
divided by the square root of the denominator,
so the given expression is equal to the square root of 2x
divided by the square root of 3y.
Now for the next step, to rationalize the denominator,
we need the denominator to simplify perfectly.
Matter of fact, now we have the square root of
one factor of three, and one factor of y.
We need the radicand to be a perfect square,
so we should be thinking, it would simplify perfectly
if we had the square root of
two factors of three and two factors of y.
Notice here, three times three, or three squared's
a perfect square, and so is y times y, or y-square.
And therefore this simplifies to
one factor of three, one factor of y, or 3y, and therefore,
if you want the denominator to simplify perfectly,
we need to multiply by the square root of 3y,
which will give us a radicand that contains
two factors of three and two factors of y,
but to create an equivalent expression,
we must also multiply the numerator by the same square root.
We multiply the numerator by the square root of 3y as well.
Now let's go ahead and multiply.
Looking at the denominator first,
we have the square root of 3y times 3y.
Let's just write that as three times three times y times y.
In the numerator, we have the square root of 2x times 3y.
Let's write that as two times three times x times y.
And now looking at the denominator,
we can now simplify because
three times three's a perfect square, and so is y times y.
So focusing on the denominator,
the square root of three times three,
or three squared is equal to one factor of three.
The square root of y times y,
or y-squared is equal to one factor of y.
The denominator is now 3y, which is considered rational,
and now, looking at the numerator,
we can see nothing simplifies here.
We still have the square root of 6xy.
The expression is now simplified
or we can say the denominator is now rationalized.
Let's look at a second example.
Here we have the square root of 5x divided by 18y,
which is equal to the square root of 5x
divided by the square root of 18y,
but notice how in this example,
we can simplify the square root of 18y,
so let's do that first.
The square root of 18y is equal to,
well 18 is equal to two times nine.
Nine's a perfect square.
Let's write this as the square root of nine times 2y,
and because the square root of nine is equal to three,
this simplifies to three square root of 2y.
So let's write the expression as the square root of 5x
divided by three square root of 2y.
And now, let's focus on rationalizing the denominator.
And again, we should be thinking,
if we had two factors of two
and two factor of y under the square root,
the square root would simplify perfectly,
and the denominator would be rationalized,
so for this example, we multiply the numerator
and the denominator by the square root of 2y.
So multiplying in the numerator,
we have the square root of 5x times 2y.
Let's write that as two times five times x times y.
In the denominator, we have three
times the square root of 2y times 2y,
which gives us two factors of two, and two factors of y,
and two times two, or two-squared's a perfect square,
and so is y times y, or y-squared, so simplifying,
we can see nothing simplifies in the numerator.
The numerator is the square root of 10xy.
In the denominator, we have three times the square root of
two times two, or two-squared is two,
and the square root of y times y or y-squared is equal to y,
and three times 2y is 6y.
The final simplified expression is
the square root of 10xy divided by 6y.
Again, this is simplified, or we can say
the denominator is now rationalized.
I hope you found this helpful.