Simplifying Square Roots That Contain Variables
Next we will simplify a square-root radical whose radicand contains a
variable.
Let’s look at these examples.
If x is a nonnegative real number, then:
since
x · x = x2 |
|
since (x3)2 = x6 |
Notice that
![](./articles_imgs/1044/simpli81.gif) |
since (x5)2 = x10 |
Notice that
![](./articles_imgs/1044/simpli83.gif) |
since
(x8)2 = x16 |
Notice that
![](./articles_imgs/1044/simpli85.gif) |
In each example, the exponent of the variable in the simplified expression
is one-half the exponent of the variable in the radicand.
If the power of x in the radicand is not a multiple of 2, we rewrite the
radicand as a product of x1 and an even power of x.
For example, let’s simplify
where
x is a nonnegative real number.
|
![](./articles_imgs/1044/simpli87.gif) |
Write x37 as x36
· x1. |
![](./articles_imgs/1044/simpli88.gif) |
Write
as the product of two radicals.
|
![](./articles_imgs/1044/simpli90.gif) |
Simplify. |
![](./articles_imgs/1044/simpli91.gif) |
In the remainder of this Topic, we will assume that each variable under a
radical represents a nonnegative real number.Be careful:
![](./articles_imgs/1044/simpli92.gif)
If x is negative, then
For example, if x = -3:
![](./articles_imgs/1044/simpli94.gif)
|