Simplifying Cube Roots That Contain Variables
Next we will simplify a cuberoot radical whose radicand contains a
variable.
We will use a procedure similar to the one we used to simplify squareroot
radicals. However, when we simplify a cuberoot 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 Â· x Â· x = x^{3} 

since
x^{4} Â· x^{4}
Â· x^{4} = x^{12} 
Notice that

since
x^{9} Â· x^{9}
Â· x^{9} = x^{27} 
Notice that

In each example, the exponent of the variable in the simplified expression
is onethird 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 x^{1} or x^{2}.
For example, letâ€™s simplify


Write x^{14} as x^{12}
Â· x^{2}. 

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,
