How can I use prime factorization to find a cube root?

Question

This is based on a lesson at Khan Academy that I didn't understand.

In the lesson, the instructor uses the number 512 as an example and the entire prime factorization consists of three groups of three 2s. However, not every number has such a nice prime factorization and I don't understand how to use prime factorization to determine a cube root.

For example, if the number is 1000, I know the cube root is 10, but how do I get there using prime factorization.

The prime factorization of 1000 is 2 * 2 * 2 * 5 * 5 * 5. Since it's six prime numbers, I can't get 3 equal groups of the same number like I could with 512. In another forum someone said to use three groups of the smallest factor, but three 2s doesn't get me 10.

What am I missing?

Answer 1

You're missing a third factor of $2$ in your prime factorization of $1000$:

$$1000 = 2\cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 = (2^3\cdot 5^3) = (2\cdot 5)^3 = (2\times 5) \cdot (2\times 5) \cdot (2 \times 5) = 10 ^3$$

Answer 2

The prime factorization for 1000 is 2*2*2*5*5*5. So you have three groups you're trying to find.

Answer 3

What you're missing is there are indeed 3 factors of 2 in 1000. Remember, when you multiply a number by 10, you add a 0 to the end of it, so it should be easy to see that

$$1000=10\times10\times10$$