Solving Equations with Radicals and Exponents
Raising Each Side to a Power
Example 1
Raising each side to a power to eliminate radicals
Solve each equation.
a)
b)
a)


Original equation 


Cube each side. 
3x + 5 
= x  1 

2x 
= 6 

x 
= 3 

Check x = 3 in the original equation:
Note that
is a real number. The solution set is
{3}. In this example we
checked for arithmetic mistakes. There was no possibility of extraneous solutions
here because we raised each side to an odd power.
b)

= x 



Original equation 
3x + 18 
= x^{2} 



Square both sides. 
x^{2} + 3x + 18 
= x^{2} 



Simplify. 
x^{2}  3x  18 
= 0 



Subtract x^{2} from each side
to get zero on one side. 
(x  6)(x + 3) 
= 0 



Multiply each side by 1 for easier factoring. 
x  6 
= 0 



Factor. 
x 
= 0 
or 
x + 3 
= 0 
Zero factor property 

= 6 
or 
x 
= 3 

Because we squared both sides, we must check for extraneous solutions. If x
= 3
in the original equation
, we get
which is not correct. If x = 6 in the original equation, we get
which is correct. The solution set is
{6}.
