Simplifying Radicals Involving Variables
To simplify radicals involving variables, we must recognize exponential expressions
that are perfect squares, perfect cubes, and so on. The expressions
x^{2}, w^{4}, y^{8}, z^{14}, and x^{50}
are perfect squares because they are squares of variables with integral powers. Any
even power of a variable is a perfect square. If we assume the variables represent
positive numbers, we can write
Helpful hint
If you use exponential notation,
then it is clear why the
square root takes half of the
exponent:
Note that when we find the square root, the result has onehalf of the original
exponent.
Example 1
Radicals with variables
Simplify each expression. Assume all variables represent positive real numbers.
Solution
a) Use the product rule to place all perfect squares under the first radical symbol
and the remaining factors under the second:


Factor out the perfect squares. 


Product rule for radicals 





Product rule for radicals 



