We are asked to rationalize the denominators.
Here we have the cube root of 5/28.
The first step is to write the expression
as an equivalent expression in the form of
the cube root of five divided by the cube root of 28.
Now from here, to rationalize the denominator,
we need the radicand in the denominator to be a perfect cube
so that it simplifies perfectly.
To help us figure out what to multiply
the numerator and denominator by,
let's look at the prime factorization of 28.
28 is equal to four times seven, seven is prime,
and four is equal to two times two, both of which are prime.
Let's write the expression as the cube root of five
divided by the cube root of two times two times seven.
Now from here, because we need
the radicand of the denominator to be a perfect cube,
we would need to have three factors of two
and three factors of seven.
The idea here is if we have the cube root
of three factors of two and three factors of seven,
the radicand will be a perfect cube,
again because we have two times two times two or two cubed,
and seven times seven times seven or seven cubed.
This cube root will simplify perfectly
to one factor of two and one factor of seven,
which is equal to 14,
and because the radicand in the denominator
only has two factors of two and one factor of seven,
we need to multiply the numerator and denominator by
the cube root of one factor of two and two factors of seven.
And of course we must also multiply the numerator
by the same cube root
in order to have an equivalent expression.
And now let's multiply.
In the numerator, we have the cube root
of five times two times seven times seven,
let's write that as two times five times seven times seven.
In the denominator, we have the cube root
of three factors of two and three factors of seven,
and notice now the radicand in the denominator
is a perfect cube,
and therefore this will simplify perfectly.
So simplifying, we have the cube root
of two times five times seven times seven which is 490.
In the denominator, the cube root of two times two times two
is equal to a factor of two,
and the cube root of seven times seven times seven
or seven cubed is equal to a factor of seven.
The final simplified expression,
or the expression with a rationalized denominator,
is the cube root of 490 divided by 14.
And again we know the cube root of 490 does not simplify
because here is the prime factorization,
and we don't have three of any factors.
Looking at our second example,
the first step is to write the fourth root of the fraction
as the fourth root of five divided by the fourth root of 12.
The next step is to look at the prime factorization of 12,
12 is equal to two times two times three,
so let's write this as the fourth root of five
divided by the fourth root of two times two times three.
Now we need to determine what to multiply
the numerator and denominator by
to make the radicand in the denominator
a perfect fourth root,
with a radicand that only has factors
raised to the fourth power.
So again we should be thinking
that if we have the fourth root
of two times two times two times two
times three times three times three times three,
this will simplify perfectly and rationalize the denominator
because we have two to the fourth here,
and three to the fourth here,
and therefore this simplifies
to one factor of two and one factor of three.
Right now we only have two factors of two
and one factor of three,
which means you need to multiply by the fourth root
of two factors of two and three factors of three,
and we must multiply the numerator and denominator
by the same fourth root.
And here we need to have two more factors of two
and three more factors of three,
so that the radicand in the denominator
is a perfect fourth root,
where it only contains factors raised to the fourth power.
And now multiplying, in the numerator,
we have the fourth root of a factor of five,
two factors of two and three factors of three,
let's write that as two times two
times three times three times three times five.
In the denominator we now have the fourth root
of four factors of two and four factors of three.
In the numerator we have the fourth root
of two times two times three times three
times three times five, which is equal to 540,
so we have the fourth root of 540,
which we know does not simplify
because we don't have four equal factors here.
And this is all divided by
the fourth root of two to the fourth,
which is equal to one factor of two
and the fourth root of three to the fourth,
which is equal to one factor of three,
and therefore the final expression simplifies
to the fourth root of 540 divided by six.
So this is the simplified expression,
or we can say the expression
with a rationalized denominator.
I hope you found this helpful.