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Power of a Power Property of Exponents
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Solving Equations with Radicals and Exponents
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Using Intercepts for Graphing Linear Equations
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Multiplying Fractions
Solving Linear Equations Containing Fractions
Evaluating Polynomials
Multiplication Property of Square and Cube  Roots
Writing a Fraction in Simplest Form
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The Pythagorean Theorem
Factoring The Difference of 2 Squares
Solving Polynomial Equations
Roots and Powers
Writing Linear Equations in Standard Form
Solving Nonlinear Equations by Substitution
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The Square of a Binomial
Solving Equations
Adding and Subtracting Like Fractions
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Finding the Equation of an Inverse Function
Slope of a Line
Rules for Nonnegative Integral Exponents

Solving Equations with Radicals and Exponents

Raising Each Side to a Power

If we start with the equation x = 3 and square both sides, we get x2 = 9. The solution set to x2 = 9 is {-3, 3}; the solution set to the original equation is {3}. Squaring both sides of an equation might produce a nonequivalent equation that has more solutions than the original equation. We call these additional solutions extraneous solutions. However, any solution of the original must be among the solutions to the new equation.

 

Caution

When you solve an equation by raising each side to a power, you must check your answers. Raising each side to an odd power will always give an equivalent equation; raising each side to an even power might not.

 

Example

Raising each side to a power to eliminate radicals

Solve the following equation:

 

Solution

Eliminate the square root by raising each side to the power 2:

= 0 Original equation
= 5 Isolate the radical.
= 52 Square both sides.
2x - 3 = 25  
2x = 28  
x = 14  

Check by evaluating x = 14 in the original equation:

= 0
= 0
= 0
0 = 0

The solution set is {14}.