FreeAlgebra Tutorials!   Home 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 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 http: 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 http: Solving Inequalities with Fractions and Parentheses http: Multiplying Polynomials Fractions Solving Quadratic and Polynomial Equations Extraneous Solutions Fractions

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Exponents and Their Properties

Recall: 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 Â· a 0 should be the same as a n Â· 1. Indeed, it is, since a n Â· a 0 = a n + 0 = a n and a n Â· 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.

 All Right Reserved. Copyright 2005-2007