8.6 Solving Equations with Radicals
Solve radical equations by using the power rule.
Power Rule for Solving an Equation with Radicals
Objectives
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.
If both sides of an equation are raised to the same power, all
solutions of the original equation are also solutions of the new
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.
When the power rule is used to solve an equation, every solution of the new
equation must be checked in the original equation.
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CLASSROOM
EXAMPLE 1
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Slide 8.6- 2
Solve radical equations by using the power rule.
Using the Power Rule
Solve
Solving an Equation with Radicals
Solution:
Check:
Step 1 Isolate the radical. Make sure that one radical term is alone
on one side of the equation.
Step 2 Apply the power rule. Raise each side of the equation to a
power that is the same as the index of the radical.
Step 3 Solve the resulting equation; if it still contains a radical,
repeat Steps 1 and 2.
Step 4 Check all proposed solutions in the original equation.
True
Since 3 satisfies the original equation, the solution set is {3}.
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CLASSROOM
EXAMPLE 2
Slide 8.6- 3
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Slide 8.6- 4
Using the Power Rule
Objective 2
Solve
Solve radical equations that
require additional steps.
Solution:
The equation has no solution, because the square root of a real
number must be nonnegative.
The solution set is .
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Slide 8.6- 5
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CLASSROOM
EXAMPLE 3
CLASSROOM
EXAMPLE 3
Using the Power Rule (Squaring a Binomial)
Using the Power Rule (Squaring a Binomial) (cont’d)
Step 4 Check each proposed solution in the original equation.
Solve
Solution:
Step 1 The radical is alone on the left side of the equation.
Step 2 Square both sides.
Step 3 The new equation is quadratic, so get 0 on one side.
False
True
The solution set is {1}. The other proposed solution, –4, is extraneous.
Slide 8.6- 7
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CLASSROOM
EXAMPLE 4
CLASSROOM
EXAMPLE 4
Using the Power Rule (Squaring a Binomial)
Slide 8.6- 8
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Using the Power Rule (Squaring a Binomial) (cont’d)
Step 4 Check each proposed solution in the original equation.
Solve
Solution:
Step 1 The radical is alone on the left side of the equation.
Step 2 Square both sides.
Step 3 The new equation is quadratic, so get 0 on one side.
False
True
The solution set of the original equation is {0}.
Slide 8.6- 9
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CLASSROOM
EXAMPLE 5
CLASSROOM
EXAMPLE 5
Using the Power Rule (Squaring Twice)
Slide 8.6- 10
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Using the Power Rule (Squaring Twice) (cont’d)
Solve
Check: x = 3
Solution:
Square both sides.
Isolate the remaining
radical.
Square both sides.
Check: x =  1
False
Apply the exponent
rule (ab)2 = a2b2.
The solution set is {1}.
True
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Slide 8.6- 11
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Slide 8.6- 12
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CLASSROOM
EXAMPLE 6
Objective 3
Using the Power Rule for a Power Greater Than 2
Solve
Solve radical equations with
indexes greater than 2.
Solution:
Cube both sides.
Check:
True
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Slide 8.6- 13
CLASSROOM
EXAMPLE 7
Objective 4
Slide 8.6- 14
Solving a Formula from Electronics for a Variable
Solve the formula for R.
Use the power rule to solve a
formula for a specified variable.
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The solution set is {9}.
Solution:
Slide 8.6- 15
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Slide 8.6- 16
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