Video Transcript
Which of the following is
equivalent to 64 to the five-sixths power?
There’s something that we need to
remember. When it comes to a fractional
power, we can rewrite this as a power to a power. What do I mean by that? Well 𝑥 to the 𝑎 times 𝑏 power,
𝑥 to the 𝑎𝑏, is the same thing as taking 𝑥 to the 𝑎 power to the 𝑏 power. When we take a power of a power, we
multiply the two exponents. And so I want to do that here. I want to write five-sixths as the
product of two different values. That could look like this: 64 to
the fifth power to the one-sixth power because one-sixth times five equals
five-sixths. But this is not the only way to
write this as the product of two factors.
We can also say 64 to the one-sixth
power to the fifth power. And again our power to a power
would mean multiplying five times one-sixth. Neither of these options show up in
our five answer choices, so we’re going to need to take another step. We have to consider another way to
write the one-sixth power. The one-sixth power is the sixth
root, and we write it with a radical sign and a six. Taking something to the one-sixth
power is the same thing as taking the sixth root. In our first case, we’re taking 64
to the fifth power, and that part doesn’t change, and then we’re taking the sixth
root of whatever 64 to the fifth power is. In our other option, we’ll first
take the sixth root of 64, and whatever we find is then taken to the fifth
power.
These three expressions are
equivalent, and only one of them is found in the answer choices. Choice e is taking the sixth root
of 64 and then taking that value to the fifth power. We can also point out the errors in
answer choices a through d. We’re not taking the fifth root of
anything, so a and c are not viable options. There would be no division in this
expression. And expression d is taking the
square root of 64 to the five- sixths power. The square root is the one-half
power, and we’ve hardly said that a power to a power means multiply. So d is saying 64 to the five
twelfths power, which is not what we started with 64 to the five-sixths power. And that leaves us with e to be our
only valid equivalent answer to 64 to the five-sixths power.
