### Video Transcript

Simplify the cube root of 64 times
π cubed.

In order to be able to simplify an
expression involving an πth root, where here π is equal to three, weβll recall one
of the properties that applies to πth roots. This property tells us what happens
when we multiply a pair of πth roots. Specifically, for positive real
numbers π and π, the πth root of π times the πth root of π is equivalent to
the πth root of ππ. Weβre going to apply this property
in reverse. And it allows us to separate the
cube root of 64π cubed into the product of the cube root of 64 and the cube root of
π cubed.

The next property weβre interested
in tells us that if π is an odd integer, which it is here, itβs three, then the
πth root of π all raised to the πth power is equal to the πth root of π to the
πth power, which is simply equal to π. And this is great. This allows us to simplify this
part of the expression, the cube root of π cubed. Since the root is odd, in other
words, π is equal to three, we can say that the cube root of π cubed is simply
equal to π. And of course, we know the value of
the cube root of 64. Itβs simply four. So we can substitute the cube root
of π cubed equals π and the cube root of 64 equals four back into our earlier
equation. And that will allow us to simplify
the original expression.

When we do, we find that the cube
root of 64 times the cube root of π cubed is four times π, which can of course be
fully simplified to four π. And so, weβve simplified the cube
root of 64π cubed. Itβs four π.