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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The properties of whole number exponents also hold for negative exponents. This table contains an example of each property.

Property Positive Integer Exponents Negative Integer Exponents
Multiplication  32 · 34 = 32 + 4 = 36
Power of a Power (52)3 = 52 · 3 = 56 (5 -2) -3 = 5(-2) · (-3) = 56
Power of a Product (5 · 7)3 = 53 · 73
Power of a Quotient

Now we will find two additional properties of negative exponents.

We’ll begin by simplifying
We apply the definition of a negative exponent.
Rewrite the division using ÷.
To divide by a fraction, multiply by its reciprocal.
Multiply the numerators. Multiply the denominators.


Notice that the bases, 2 and 5, have moved to the opposite side of the division bar, and the signs of the their exponents changed.

This turns out to be true in general.

Next, we’ll use this relationship to rewrite a quotient raised to a negative power.

For example, we’ll simplify
We use the Power of a Quotient Property.
As in the previous example, we move each base to the opposite side of the division bar and change the sign of each exponent.
Again, we use the Power of a Quotient Property.

We see that

Notice that the new base, , is the reciprocal of the original base. Also notice the new exponent, 2, is the opposite of the original exponent.


To rewrite a fraction raised to a negative power, just “flip” the fraction and change the negative power to positive.