5.2 Exponent Properties Involving Quotients


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1 5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use the Quotient of Powers Property You sw in the lst section tht we cn use eponent rules to simplify products of numbers nd vribles. In this section, you will lern tht there re similr rules you cn use to simplify quotients. Lets tke n emple of quotient, 7 divided by. 7 You should see tht when we divide two powers of, the number of fctors of in the solution is the difference between the fctors in the numertor of the frction, nd the fctors in the denomintor. In other words, when dividing epressions with the sme bse, keep the bse nd subtrct the eponent in the denomintor from the eponent in the numertor. Quotient Rule for Eponents: n m n m When we hve problems with different bses, we pply the quotient rule seprtely for ech bse. 5 y y y y y y y y OR 5 y 5 y y Emple Simplify ech of the following epressions using the quotient rule. 0 5 b 6 c 5 b 4 b Apply the quotient rule b c 5 b 4 b 5 b 4 b Now lets see wht hppens if the eponent on the denomintor is bigger thn the eponent in the numertor.
2 Emple Divide Apply the quotient rule. A negtive eponent!? Wht does tht men? Lets do the division longhnd by writing ech term in fctored form. 6 We see tht when the eponent in the denomintor is bigger thn the eponent in the numertor, we still subtrct the powers. This time we subtrct the smller power from the bigger power nd we leve the s in the denomintor. When you simplify quotients, to get nswers with positive eponents you subtrct the smller eponent from the bigger eponent nd leve the vrible where the bigger power ws. We lso discovered wht negtive power mens. We ll lern more on this in the net section! Emple Simplify the following epressions, leving ll powers positive. 6 b b 6 5 b Subtrct the eponent in the numertor from the eponent in the denomintor nd leve the s in the denomintor. b Apply the rule on ech vrible seprtely. 6 6 b 6 5 b 5 b6 b5 The Power of Quotient Property When we pply power to quotient, we cn lern nother specil rule. Here is n emple. 4 y y y y y y y y y 8 Notice tht the power on the outside of the prenthesis multiplies with the power of the in the numertor nd the power of the y in the denomintor. This is clled the power of quotient rule. Power Rule for Quotients n p y n p m y m p
3 Simplifying Epressions Involving Quotient Properties of Eponents Lets pply the rules we just lerned to few emples. When we hve numbers with eponents nd not vribles with eponents, we evlute. Emple 4 Simplify the following epressions b 5 c In ech of the emples, we wnt to evlute the numbers. Use the quotient rule first. Then evlute the result OR We cn evlute ech prt seprtely nd then divide b Use the quotient rule first Then evlute the result OR We cn evlute ech prt seprtely nd then reduce
4 It mkes more sense to pply the quotient rule first for emples nd b. In this wy the numbers we re evluting re smller becuse they re simplified first before pplying the power. c Use the power rule for quotients first Then evlute the result OR We evlute inside the prenthesis first Then pply the power outside the prenthesis When we hve just one vrible in the epression, then we pply the rules strightforwrdly. Emple 5: Simplify the following epressions: 5 b 5 : Use the quotient rule. b Use the power rule for quotients first Then pply the quotient rule 0 5 5
5 OR Use the quotient rule inside the prenthesis first Then pply the power rule. 5 5 When we hve mi of numbers nd vribles, we pply the rules to ech number or ech vrible seprtely. Emple 6 Simplify the following epressions. 6 y b b 8 7 b We group like terms together. 6 y 6 y We reduce the numbers nd pply the quotient rule on ech grouping. b We pply the quotient rule inside the prenthesis first. y > b b 8 7 b 4 4 Apply the power rule for quotients. b 4 4 b4 6 8 In problems tht we need to pply severl rules together, we must keep in mind the order of opertions. Emple 7 Simplify the following epressions. 6
6 b 6 4b b 5 6 We pply the power rule first on the first prenthesis. Then pply the quotient rule to simplify the frction Apply the product rule to simplify b Simplify inside the first prenthesis by reducing the numbers. 4 Then we cn pply the power rule on the first prenthesis. b 5 b 6 Group like terms together. 4 b 5 b b 5 b 6 Apply the quotient rule on ech frction b 5 b b 6 b b 64 b5 0 b Review Questions Evlute the following epressions
7 Simplify the following epressions b 4 b 8. 6 y y 6 0y 5 6 y 4 6 4b 4 5b b 4 b 4 6 b 4 bc 6bc 4b c Review Answers b 9 8. y y 4. 6 y b b c 4
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