Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
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
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
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
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
Quadratic Functions
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
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
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Exponents and Their Properties


if n is a positive integer.

New Stuff:

When you have a product of two terms with the same base, the exponents add together.

Rule (The Product Rule)

For any number a and positive integers m and n , a m a n = a m + n.

  • Multiplication of powers leads to addition of the exponents. When you raise something to a power, you are really multiplying it by itself several times. Thus, if you raise a power to a power, you should add the exponent to itself several times. In other words, multiply the powers.

Rule (Power Rule)

For any number a and any positive integers m and n , ( a m ) n .

  • Since multiplication leads to addition of exponents, it makes sense that division would lead to subtraction.

Rule (The Quotient Rule)

For any number a ( a 0) and any positive integers m and n for which m > n ,.

The quotient rule allows us to determine what a 0 exponent should mean. Suppose that we allow m = n in the rule above. Then we have (since the top and bottom are the same). But we also have . Thus, we should define a 0 = 1.

Definition ( 0 as an Exponent)

For any real number a , a 0, a 0 = 1.

  • That this is a good way to define 0 as an exponent is verified by looking at how well this works with the other rules.

The multiplication rule: If this is a good definition, then a n · a 0 should be the same as a n · 1. Indeed, it is, since a n · a 0 = a n + 0 = a n and a n · 1 = a n .

The power rule: The power rule is really just an extension of the multiplication rule, so the definition works with the power rule.

The quotient rule: We made the definition specifically so it works with the quotient rule, so this works too.

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