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
![](./articles_imgs/1016/simpli3.gif) |
since
x9 · x9
· x9 = x27 |
Notice that
![](./articles_imgs/1016/simpli5.gif) |
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:
![](./articles_imgs/1016/simpli6.gif)
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
|
![](./articles_imgs/1016/simpli8.gif) |
Write x14 as x12
· x2. |
![](./articles_imgs/1016/simpli9.gif) |
Notice that 12 is a multiple of 3.
|
|
Write as the product of two radicals.
|
![](./articles_imgs/1016/simpli10.gif) |
Simplify. |
![](./articles_imgs/1016/simpli11.gif) |
So,
![](./articles_imgs/1016/simpli12.gif)
Example
Simplify:
![](./articles_imgs/1016/simpli13.gif)
Solution |
![](./articles_imgs/1016/simpli14.gif) |
Factor the radicand, using perfect
cube factors when possible.
|
![](./articles_imgs/1016/simpli15.gif) |
Write as a product of four radicals.
|
![](./articles_imgs/1016/simpli16.gif) |
Simplify the cube root of each perfect
cube. |
![](./articles_imgs/1016/simpli17.gif) |
Combine the remaining radicals. |
![](./articles_imgs/1016/simpli18.gif) |
So,
![](./articles_imgs/1016/simpli19.gif)
|