Simplifying Expressions Involving Variables
When simplifying expressions involving rational exponents and variables, we must
be careful to write equivalent expressions. For example, in the equation
(x^{2})^{1/2} = x it looks as if we are correctly applying the power of a power rule. However, this
statement is false if x is negative because the 1/2 power on the lefthand side
indicates the positive square root of x^{2}. For example, if x = 3, we get
[(3)^{2}]^{1/2} = 9^{1/2} = 3, which is not equal to 3. To write a simpler equivalent expression for (x^{2})^{1/2}, we
use absolute value as follows.
Square Root of x^{2
}
(x^{2})^{1/2} =  x
 for any real number x.
Note that the equation (x^{2})^{1/2} =
 x  is an identity. It is also necessary to use
absolute value when writing identities for other even roots of expressions involving
variables.
Example 1
Using absolute value symbols with exponents
Simplify each expression. Assume the variables represent any real numbers and use
absolute value symbols as necessary.
a) (x^{8}y^{4})^{1/4 }
b)
Solution
a) Apply the power of a product rule to get the equation (x^{8}y^{4})^{1/4}
= x^{2}y. The lefthand
side is nonnegative for any choices of x and y, but the righthand side is
negative when y is negative. So for any real values of x and y we have (x^{8}y^{4})^{1/4}
= x^{2} y
.
b) Using the power of a quotient rule, we get
This equation is valid for every real number x, so no absolute value signs are
used.
Helpful hint
We usually think of squaring
and taking a square root as inverse
functions, which they
are as long as we stick to positive
numbers.We can square 3
to get 9, and then find the
square root of 9 to get 3â€”
what we started with. We
donâ€™t get back to where we
began if we start with 3.
Because there are no real even roots of negative numbers, the expressions
a^{1/2}, x^{3/4}, and y^{1/6}
are not real numbers if the variables have negative values. To simplify matters, we
sometimes assume the variables represent only positive numbers when we are
working with expressions involving variables with rational exponents. That way we
do not have to be concerned with undefined expressions and absolute value.
Example 2
Expressions involving variables with rational exponents
Use the rules of exponents to simplify the following. Write your answers with
positive exponents. Assume all variables represent positive real numbers.
Solution
a) x^{2/3}x^{4/3} 
= x^{6/3 } 
Use the product rule to add the exponents. 

= x^{2} 
Reduce the exponent. 
b)

= a^{1/2  1/4} 
Use the quotient rule to subtract the exponents. 

= a^{1/4} 
Simplify. 
c) (x^{1/2}y^{ 3})^{1/2} 
= (x^{1/2})^{1/2}(y^{ 3})^{1/2} 
Power of a product rule 

= x^{1/4}y^{ 3/2} 
Power of a power rule 


Definition of negative exponent 
d) Because this expression is a negative power of a quotient, we can first find the
reciprocal of the quotient, then apply the power of a power rule:
