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Adding and Subtracting Fractions with Like Denominators
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Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
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Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
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Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
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Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
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Exponent Laws
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Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
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The parabola
Computations with Scientific Notation
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Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
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The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
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Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
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Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
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Solving Equations
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Solving Equations
Product and Quotient of Functions
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Positive Rational Exponents

When a rational number is used as an exponent, the denominator indicates “root” and the numerator indicates “power.” For example, 82/3 means (81/3)2. We take the cube root of 8 to get 2, then square 2 to get 4. Thus 82/3 = 4.

In general, am/n is the mth power of the nth root of a. Rational Exponents If m and n are positive integers, then am/n = (a1/n)m, provided that a1/n is defined.


Example 1

Evaluating expressions with positive rational exponents

Evaluate each expression.

a) 163/4

b) 274/3

c) (-8)2/3

d) -43/2


a) To evaluate 163/4, take the fourth root of 16 to get 2 and then cube 2 to get 8. These steps are written as follows:

163/4 = (161/4)3 = 23 = 8

b) To evaluate 274/3, take the cube root of 27 to get 3, and then raise 3 to the fourth power to get 81:

274/3 = (271/3)4 = 34 = 81

c) (-8)2/3 = [(-8)1/3]2 = (-2)2 = 4

d) -43/2 = -(41/2)3 = -23 = -8


We can evaluate -43/2 because the negative sign is saved until last. But (-4)3/2 is not a real number because (-4)1/2 is not a real number.

By the definition of rational exponents we find the root and then the power. For example,

82/3 = (81/3)2 = 22 = 4.

However, we get the same result if we find the power and then the root:

82/3 = (82)1/3 = 641/3 = 4

In general, am/n is also the nth root of the mth power of a.


Evaluating am/n in Either Order

If m and n are positive integers, then am/n = (a1/n)m = (am)1/n, provided that a1/n is defined.


The fact that the power can be found before the root will be used later in this chapter. It is not useful in simply evaluating expressions because if the power is evaluated first, we might have to find the root of a very large number. For example, finding the power first in Example 1(b), we get

274/3 = (274)1/3 = 531,4411/3

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It does not matter whether you use a fraction or a decimal as the exponent as long as the decimal is the exact value of the fraction. You will get the same result whether you find the root first or the power first.

Helpful hint

Note that in am/n we do not require m/n to be reduced. As long as the nth root of a is real, then the value of am/n is the same whether or not m/n is in its lowest terms.

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