Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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In this section we will extend the definition of exponents to include all integers and to learn some rules for working with integral exponents.

Positive and Negative Exponents

Positive integral exponents provide a convenient way to write repeated multiplication or very large numbers. For example,

2 · 2 · 2 = 23, y · y · y · y = y4, and 1,000,000,000 = 109.

We refer to 23 as “2 cubed,” “2 raised to the third power,” or “a power of 2.”

Positive Integral Exponents

If a is a nonzero real number and n is a positive integer, then

In the exponential expression an, the base is a, and the exponent is n.

We use 2-3 to represent the reciprocal of 23. Because 23 = 8, we have . In general, a-n is defined as the reciprocal of an.

Negative Integral Exponents

If a is a nonzero real number and n is a positive integer, then

(If n is positive, -n is negative.)

To evaluate 2-3, you can first cube 2 to get 8 and then find the reciprocal to get , or you can first find the reciprocal of 2 (which is ) and then cube to get . So

The power and the reciprocal can be found in either order. If the exponent is -1, we simply find the reciprocal. For example,

Because 23 and 2-3 are reciprocals of each other, we have

These examples illustrate the following rules.

Rules for Negative Exponents

If a is a nonzero real number and n is a positive integer, then

Example 1

Negative exponents

Evaluate each expression.

a) 3-2

b) (-3)-2

c) -3-2




Definition of negative exponent
Definition of negative exponent
Evaluate 3-2, then take the opposite.
The reciprocal of .

The cube of .

The reciprocal of 5-3 is 53.


We evaluate -32 by squaring 3 first and then taking the opposite. So -32 = -9, whereas (-3)2 = 9. The same agreement also holds for negative exponents. That iswhy the answer to Example 1(c) is negative.