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