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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 Odd-Root Property

Because (-2)3 = -8 and 23 = 8, the equation x3 = 8 is equivalent to x = 2. The equation x3 = -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.

 

Odd-Root Property

If n is an odd positive integer, xn = k is equivalent to for any real number k.

 

Example 1

Using the odd-root property

Solve each equation.

a) x3 = 27

b) x5 + 32 = 0

c) (x - 2)3 = 24

Solution

a) x3 = 27  
x Odd-root property
x = 3  

Check 3 in the original equation. The solution set is {3}.

b) x5 + 32 = 0  
x5 = -32 Isolate the variable.
x Odd-root property
x = -2  

Check -2 in the original equation. The solution set is {-2}.

c) (x - 2)3 = 24 Odd-root property
x + 2
x =  

Check. The solution set is { }.

 

 
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