Solving Equations with Radicals and Exponents
One of our goals in algebra is to keep increasing our knowledge of solving equations
because the solutions to equations can give us the answers to various applied
questions. In this section we will apply our knowledge of radicals and exponents to
solving some new types of equations.
The OddRoot Property
Because (2)^{3} = 8 and 2^{3} = 8, the equation x^{3}
= 8 is equivalent to x = 2. The
equation x^{3} = 8 is equivalent to x = 2. Because there is only one real odd root
of each real number, there is a simple rule for writing an equivalent equation in this
situation.
OddRoot Property
If n is an odd positive integer,
x^{n} = k is equivalent to
for any real number k.
Example 1
Using the oddroot property
Solve each equation.
a) x^{3} = 27
b) x^{5} + 32 = 0
c) (x  2)^{3} = 24
Solution
a) x^{3} 
= 27 

x 

Oddroot property 
x 
= 3 

Check 3 in the original equation. The solution set is {3}.
b) x^{5} + 32 
= 0 

x^{5} 
= 32 
Isolate the variable. 
x 

Oddroot property 
x 
= 2 

Check 2 in the original equation. The solution set is {2}.
c) (x  2)^{3} 
= 24 
Oddroot property 
x + 2 


x 
=


Check. The solution set is {
}.
