Video: Expressing a Number with a Rational Exponent in Radical Form

Which of the following is equivalent to 64^(5/6)? [A] the fifth root of 6⁶⁴ [B] 64⁵/the sixth root of 64 [C] the fifth root of 64⁶ [D] the square root of 64^(5/6) [E] (the sixth root of 64)⁵

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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.

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