Exponents and Polynomials - XYZ Custom Plus

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590Chapter 9 Exponents and PolynomialsThe properties of exponents we have developed so far hold for negative exponentsas well as positive exponents. For example, Property 1, our multiplicationproperty for exponents, is still written asa r ⋅ a s = a r+sbut now r and s can be negative numbers also.4. Simplify: 3 4 ⋅ 3 −7Example 4Simplify: 2 5 ⋅ 2 −7SolutionThis is multiplication with the same base, so we add exponents.2 5 ⋅ 2 −7 = 2 5+(−7) Property 1 for exponents= 2 −2 Addition= _122 Definition of negative exponents= _1 The square of 2 is 44When we simplify expressions containing negative exponents, let’s agree that thefinal expression contains only positive exponents.5. Simplify: x 6 ⋅ x −10Example 5Simplify: x 9 ⋅ x −12Solution Again, because we have the product of two expressions with thesame base, we use Property 1 for exponents to add exponents.Bx 9 ⋅ x −12 = x 9+(−12) Property 1 for exponents= x −3 Add exponents= 1 _x 3Division with ExponentsDefinition of negative exponentsTo develop our next property of exponents, we use the definition for positive exponents.Consider the expression _x6. We can simplify by expanding the numerator and4xdenominator and then reducing to lowest terms by dividing out common factors._x 6x = x __⋅ x ⋅ x ⋅ x ⋅ x ⋅ x 4 x ⋅ x ⋅ x ⋅ xExpand numerator and denominatorx∙ ⋅ x∙ ⋅ x∙ ⋅ x∙ ⋅ x ⋅ x= __x∙ ⋅ x∙ ⋅ x∙ ⋅ x∙= x ⋅ x}Divide out common factors= x 2 Write answer with exponent 2Note that the exponent in the answer is the difference of the exponents in theoriginal problem. More specifically, if we subtract the exponent in the denominatorfrom the exponent in the numerator, we obtain the exponent in the answer.This discussion leads us to another property of exponents.Answers4. 1 _27 5. 1 _x 4
9.4 Negative Exponents591Division Property for Expressions Containing ExponentsIf a is any number other than zero, and r and s are integers, then_ ar= ar−ssaIn words: To divide two numbers with the same base, subtract the exponent inthe denominator from the exponent in the numerator, and use the commonbase as the base in the answer.Example 6Simplify: _252 8Solution Using our new property, we subtract the exponent in the denomina-6. Simplify: _262 8tor from the exponent in the numerator. The result is an expression containing anegative exponent._ 252 = 8 25−8 Division property for exponents= 2 −3 Subtraction= _123 Definition of negative exponents= _1 The cube of 2 is 88To further justify our new property of exponents, we can rework the problemshown in Example 6, without using the division property for exponents, to seethat we obtain the same answer._252 = ___2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 Expand numerator and denominator8 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 22∙ ⋅ 2∙ ⋅ 2∙ ⋅ 2∙ ⋅ 2∙= __2∙ ⋅ 2∙ ⋅ 2∙ ⋅ 2∙ ⋅ 2∙ ⋅ 2 ⋅ 2 ⋅ 2 }Divide out common factors= 1_2 ⋅ 2 ⋅ 2 = _1 8 MultiplyAs you can see, the answer matches the answer obtained by using the divisionproperty for exponents.Example 7Divide: 10 −3 ÷ 10 57. Divide: 10 −5 ÷ 10 3Solution We begin by writing the division problem in fractional form. Thenwe apply our division property.10 −3 ÷ 10 5 = _10−3 Write problem in fractional form510= 10 −3−5 Division property for exponents= 10 −8 Subtraction= _1108 Definition of negative exponentsWe can leave the answer in exponential form, as it is, or we can expand thedenominator to obtain1Answers= __100,000,000 6. 1 _4 7. 1 _10 8
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