### Video Transcript

Simplify the cube root of the cube
root of 512 over π₯ to the power of 63.

Here we have a rather interesting
problem which involves two different cube roots. The way that we solve this problem
is by simplifying the innermost cube root first. So we simplify the cube root of 512
over π₯ to the power 63. To do this, weβll need to apply
some exponent rules. The first rule we can apply is that
for any real values of the πth root of π and the πth root of π, then the πth
root of π over the πth root of π is equivalent to the πth root of π over
π.

And so we know that the cube root
of 512 over π₯ to the power 63 is equal to the cube root of 512 over the cube root
of π₯ to the 63rd power. We can then simplify the numerator
and denominator in turn. So letβs work out the cube root of
512. Remember that to work out the cube
root of 512, that means that weβre looking for the value which when written three
times and multiplied gives us an answer of 512. That would be eight. And so the numerator simplifies to
eight.

Next, letβs see how we simplify the
cube root of π₯ to the 63rd power. To do this, we can use another
exponent rule. That is that the πth root of π is
equal to π to the power of one over π. The cube root here can therefore be
written as the power of one-third. But we can simplify this
denominator π₯ to the power of 63 to the power of one-third even further. We remember that π to the power π₯
to the power π¦ is equal to π to the power π₯π¦. Therefore, on the denominator, we
multiply the exponents of 63 and one-third. 63 times one-third is 21, leaving
us with π₯ to the power 21 on the denominator.

Remember, at this point, we havenβt
fully simplified the question. We have only simplified the cube
root of 512 over π₯ to the power 63 as eight over π₯ to the 21st power. We still need to take a further
cube root of this expression. In order to simplify this
expression, we can apply the first exponent rule again. Therefore, on the right-hand side,
we have the cube root of eight over the cube root of π₯ to the power 21. Taking the numerator and
denominator in turn, we work out that the cube root of eight is two.

Then to simplify the cube root of
π₯ to the power of 21, we apply the second exponent rule. We can then write this as π₯ to the
power 21 to the power one-third. Applying the third exponent rule,
we know that we can multiply the indices of 21 and one-third. And 21 times one-third is seven,
which leaves π₯ to the power seven on the denominator. We cannot simplify this any
further. So we can give the answer that the
cube root of the cube root of 512 over π₯ to the 63rd power is two over π₯ to the
seventh power.