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