Simplifying Cube Roots That Contain Variables

Next we will simplify a cube-root radical whose radicand contains a variable.

We will use a procedure similar to the one we used to simplify square-root radicals. However, when we simplify a cube-root radical, we divide the exponent of the variable by 3 (instead of 2).

Here are some examples.

If x is any real number, then:

since x · x · x = x3
since x4 · x4 · x4 = x12	Notice that
since x9 · x9 · x9 = x27	Notice that

In each example, the exponent of the variable in the simplified expression is one-third the exponent of the variable in the radicand.

Be careful:

If the power of x in the radicand is not a multiple of 3, we rewrite the radicand as a product where one of the factors has a power that is a multiple of 3 and the other factor is x¹ or x².

For example, let’s simplify
Write x14 as x12 · x2.
Notice that 12 is a multiple of 3.
Write as the product of two radicals.
Simplify.

So,

Example

Simplify:

Solution
Factor the radicand, using perfect cube factors when possible.
Write as a product of four radicals.
Simplify the cube root of each perfect cube.
Combine the remaining radicals.

So,