Video Transcript
Simplify the cube root of 64 times
𝑚 cubed.
In order to be able to simplify an
expression involving an 𝑛th root, where here 𝑛 is equal to three, we’ll recall one
of the properties that applies to 𝑛th roots. This property tells us what happens
when we multiply a pair of 𝑛th roots. Specifically, for positive real
numbers 𝑎 and 𝑏, the 𝑛th root of 𝑎 times the 𝑛th root of 𝑏 is equivalent to
the 𝑛th root of 𝑎𝑏. We’re going to apply this property
in reverse. And it allows us to separate the
cube root of 64𝑚 cubed into the product of the cube root of 64 and the cube root of
𝑚 cubed.
The next property we’re interested
in tells us that if 𝑛 is an odd integer, which it is here, it’s three, then the
𝑛th root of 𝑎 all raised to the 𝑛th power is equal to the 𝑛th root of 𝑎 to the
𝑛th power, which is simply equal to 𝑎. And this is great. This allows us to simplify this
part of the expression, the cube root of 𝑚 cubed. Since the root is odd, in other
words, 𝑛 is equal to three, we can say that the cube root of 𝑚 cubed is simply
equal to 𝑚. And of course, we know the value of
the cube root of 64. It’s simply four. So we can substitute the cube root
of 𝑚 cubed equals 𝑚 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 𝑚 cubed is four times 𝑚, which can of course be
fully simplified to four 𝑚. And so, we’ve simplified the cube
root of 64𝑚 cubed. It’s four 𝑚.
