Exponents and Polynomials - XYZ Custom Plus

Negative Exponents9.4Up to this point, all the exponents we have worked with have been positivenumbers. We want to extend the type of numbers we can use for exponents toinclude negative numbers as well. The definition that follows allows us to workwith exponents that are negative numbers.ANegative ExponentsDefinitionNegative Exponents If r is a positive integer and a is any number otherthan zero, then9.4 Negative ExponentsObjectivesA Simplify expressions containingnegative exponents.B Understand and apply the divisionproperty for exponents.Examples now playing atMathTV.com/booksindicatesexamples illustrate.Example 1The first stepdefinition.2 −3 = ​_123​ = ​_1 ​8Example 2First we5 −2 = ​_152​ = ​_125see from ourexponents to expressionsExample 3The fact thatwe use to(−3) −2 ==understanding2 1 = 22 −1 = ​_1 2 ​ that negativeSimplify: 2 −3is to rewriteAfter that, weDefinition ofThe cube ofSimplify: 5 −2write the expressionDefinition ofThe square offirst two examples,containingSimplify: (−3)the basesimplify:1_(−3) 2​ DefinitionThe squareof negative2 2 = 4 = ​_1 4 ​ a −r = ​_1 a r​exponentsthe expressionsimplify.negative exponentsis 8withnegative exponents5 is 25whennegative in this problemof negativeof −3 isexponents,2 3 = 8 = ​_1 8 ​ Because division by0 is undefined, ourdefinition includes thegive us reciprocals, as the followingrestriction a ≠ 0.Practice Problemswith a positive exponent, 1. Simplify: 2 −42. Simplify: 4 −2a positive exponent, then wewe apply the definition for negativeexponents, we end up with3. Simplify: (−2) −3is negative does not changeexponentslook over the two lines below.2 4 = 16Answers2−4 = ​_116 ​ 1. ​_1 16 −​ _ 1 8The definitionSolutionby using our2Solutionsimplify.​As you canreciprocals.Solutionthe procedure​​_1 ​99To check yourNote589