Simplifying Square Roots That Contain Variables
Next we will simplify a squareroot radical whose radicand contains a
variable.
Letâ€™s look at these examples.
If x is a nonnegative real number, then:
since
x Â· x = x^{2} 

since (x^{3})^{2} = x^{6} 
Notice that

since (x^{5})^{2} = x^{10} 
Notice that

since
(x^{8})^{2} = x^{16} 
Notice that

In each example, the exponent of the variable in the simplified expression
is onehalf 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 x^{1} and an even power of x.
For example, letâ€™s simplify
where
x is a nonnegative real number.


Write x^{37} as x^{36}
Â· x^{1}. 

Write
as the product of two radicals.


Simplify. 

In the remainder of this Topic, we will assume that each variable under a
radical represents a nonnegative real number.Be careful:
If x is negative, then
For example, if x = 3:
