Solving Equations with a Fractional Exponent
An expression that contains a fractional exponent can be written using a
radical. Fractional exponents are often referred to as rational exponents.
Definition â€” Rational Exponent
If a is a real number, and m and n are natural numbers, then:
Here we assume that if n is even then a ≥
0.
Here are two examples:
Notice that in a^{m/n} the number in the denominator of the fraction, n,
becomes the index in the radical,
Example
Solve for x: (2x  6)^{2/3}  7 = 3
Solution 
(2x  6)^{2/3}  7 
= 3 
First, rewrite (2x  6)^{2/3} as a radical. 

= 3 
Step 1 Isolate a radical term.
Add 7 to both sides.
Step 2 Apply the Principle of Powers. 

= 4 
Cube each side of the equation. 

= (4)^{3} 
Step 3 Solve the resulting equation.
Simplify.
Write the left side as a product.
Simplify.

(2x  6)^{2}
(2x  6)(2x  6)
4x^{2}  24x + 36 
= 64
= 64
= 64 
Write in standard form.
Divide both sides by 4.
Factor.
Use the Zero Product Property.
Solve for x.
Step 4 Check the solution.

4x^{2}  24x  28 x^{2}  6x  7
(x  7)(x + 1)
x  7 = 0 or x + 1
x = 7 or x 
= 0 = 0
= 0
= 0
= 1 
We leave the check for you (both solutions check).
So, the solutions are x = 7 and x = 1.
