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Evaluate the cube root of 27.
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In order to evaluate the cube root of 27, we want to break down 27 into its prime factors.
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And we can do so using the branch method.
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Think of it as three branches.
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So 27 breaks out into what number times what number?
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Nine times three would work.
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So what about nine or three?
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Do they break down?
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Well, nine is the same thing as three times three.
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Now, what about this three?
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Is it prime or does it break down?
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Three is only divisible by itself and one, which means it’s prime.
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So it doesn’t break down anymore.
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So we need to look at the end of every branch.
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So we have three, another three, and another three.
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So we could rewrite 27 as three times three times three.
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However, three times three times three can be rewritten as three cubed.
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And when we cube root something cubed, the cube root and the cubed disappear.
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So our answer will be three.
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So why is it that the cube root will cancel the cubed?
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So let’s look at the cube root of any number cubed.
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We’ll call that number 𝑎.
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And we can rewrite the cube root as a one-third power.
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So notice the three and the one-third are right next to each other.
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So this means we need to multiply them.
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In order to multiply these, we multiply the numerators together and we multiply the denominators together.
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The denominator of the three is a one.
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So for our numerators, the numbers on top we’ve three times one which is three.
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And on the denominator, the bottom, we have one times three which is three and three over three is equal to one.
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So we’ve eight to the first power and anything to the first power is just itself.
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So like we said, if we cube root a number that’s cubed, the cube root and the cubed disappear and it’s just that number.
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So once again, our answer is three.