468 CHAPTER 8 Roots, Radicals, and Root Functions
OBJECTIVES An equation that includes one or more radical expressions with a variable is called a
radical equation.
2 x- 4 = 8, 25 x+ 12 = 322 x- 1 ,and 236 + x= 27
Solving Equations with Radicals
8.6
1 Solve radical
equations by using
the power rule.
2 Solve radical
equations that
require additional
steps.
3 Solve radical
equations with
indexes greater
than 2.
4 Use the power rule
to solve a formula
for a specified
variable.OBJECTIVE 1 Solve radical equations by using the power rule.The equa-
tion has only one solution. Its solution set is. If we square both sides of
this equation, we get This new equation has twosolutions: and 1. Notice
that the solution of the original equation is also a solution of the equation following
squaring. However, that equation has another solution, that is nota solution of
the original equation.
When solving equations with radicals, we use this idea of raising both sides to a
power. It is an application of the power rule.
- 1,
x^2 =1. - 1
x= 1 516
Examples of
radical equationsPower Rule for Solving an Equation with RadicalsIf both sides of an equation are raised to the same power, all solutions of the orig-
inal equation are also solutions of the new equation.
CAUTION When the power rule is used to solve an equation, every solution of
the new equation must be checked in the original equation.
The power rule does not say that all solutions of the new equation are solutions
of the original equation. They may or may not be.Solutions that do not satisfy the
original equation are called extraneous solutions.They must be rejected.
Using the Power RuleSolve
Apply the exponents.
Subtract 4.
Divide by 3.CHECK Original equation
Let.Simplify.8 = 8 ✓ True
264 8
23 # 20 + 4 8 x= 20
23 x+ 4 = 8
x= 20
3 x= 60
3 x+ 4 = 64
(^) A 23 x+ (^4) B
2
= 82
23 x+ 4 =8.
EXAMPLE 1
Use the power rule
and square each side.Since 20 satisfies the originalequation, the solution set is 5206. NOW TRY
NOW TRY
EXERCISE 1Solve 29 x+ 7 =5.
NOW TRY ANSWER
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