How to Factor Cubed Roots

By Isaiah David

To simplify a cubed root, you have to factor it. Factoring a cube root is like factoring any other number. The difference is that you have to find numbers that are cubed to remove them from the radical sign. Fortunately, there are not many numbers that can be cubed without becoming very large. That means that, generally, when you have to factor a cube root in school, you are dealing with small factors.

Factor out 2s. If the cube root is even, keep factoring out the number 2 until the number is odd. For example, for the cubed root of 40, we get: 40 = 2 x 20 = 2 x 2 x 10 = 2 x 2 x 2 x 5

Factor out 3s. You can tell if a number has the number 3 as a factor by adding up the integers in it and seeing if they add up to a power of 3. For example, 15 is divisible by 3 because 1 + 5 = 6, which is a 3. As in Step 1, keep factoring out 3 until you can't factor out any more: 54 = 2 x 27 = 2 x 3 x 9 = 2 x 3 x 3 x 3

Factor out the 5s. You can tell a number has 5 as a factor because it ends with 0 or 5.

Factor out the 7s. Unfortunately, there is no clear pattern to 7s. You will either have to memorize your 7 multiplication table or experimentally divide the number to see if it divides evenly.

After you completely factor the cubed root, move any number that goes in 3 times to the left side of the radical. For example: ³?8 = ³?2 x 2 x 2 = 2

Multiply numbers that you can't remove from the radical sign to get the final form of the cubed root: ³?120 = ³?2 x 60 = ³?2 x 2 x 30 = ³?2 x 2 x 2 x 15 = ³?2 x 2 x 2 x 3 x 5 = 2³?3 x 5 = 2³?15

Tip

Occasionally, you will get cubed roots with 11, 13, 17 or other high prime numbers in them. There is no straightforward method for factoring out high primes. You will just have to guess and check.

Warning

It is easy to get cubed roots and square roots confused. Remember, in square roots you factor out numbers that are squared under the radical sign, but in cubed roots, you factor out numbers that are cubed.

About the Author

Isaiah David is a freelance writer and musician living in Portland, Ore. He has over five years experience as a professional writer and has been published on various online outlets. He holds a degree in creative writing from the University of Michigan.