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Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
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Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Powers
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
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Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Fractions
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Adding
Quadratic Functions
Conjugates
Factoring
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Percents
Arithmetics
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Polynomials
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
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Solving Inequalities with Fractions and Parentheses
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Multiplying Polynomials
Fractions
Solving Quadratic and Polynomial Equations
Extraneous Solutions
Fractions
 

Equations Involving Rational Exponents

Equations involving rational exponents can be solved by combining the methods that you just learned for eliminating radicals and integral exponents. For equations involving rational exponents, always eliminate the root first and the power second.

 

Example 1

Eliminating the root, then the power

Solve each equation.

a) x2/3 =4

b) (w - 1)-2/5 = 4

Solution

a) Because the exponent 2/3 indicates a cube root, raise each side to the power 3:

x2/3 = 4 Original equation
(x2/3)3 = 43 Cube each side.
x2 = 64 Multiply the exponents:
x = 8 or x = -8 Even-root property

All of the equations are equivalent. Check 8 and -8 in the original equation. The solution set is {-8, 8}.

= 4 Original equation
[(w - 1)-2/5]-5 = 4-5 Raise each side to the power -5 to eliminate the negative exponent.
(w - 1)2 Multiply the exponents:
w - 1 Even-root property
w - 1 or w - 1
w or w

 

Check the values in the original equation.The solution set is .

An equation with a rational exponent might not have a real solution because all even powers of real numbers are nonnegative.

Helpful hint

Note how we eliminate the root first by raising each side to an integer power, and then apply the even-root property to get two solutions in Example 1(a). A common mistake is to raise each side to the 3/2 power and get x = 43/2 = 8. If you do not use the even-root property you can easily miss the solution -8.

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