Apply Properties of Rational Exponents. The properties of integer exponents you learned in Lesson 5.1 can also be applied to rational exponents.

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1 TEKS 6. 1, A..A Appl Properties of Ratioal Epoets Before You simplified epressios ivolvig iteger epoets. Now You will simplif epressios ivolvig ratioal epoets. Wh? So ou ca fid velocities, as i E. 8. Ke Vocabular simplest form of a radical like radicals The properties of iteger epoets ou leared i Lesso.1 ca also be applied to ratioal epoets. KEY CONCEPT For Your Notebook Properties of Ratioal Epoets Let a ad b be real umbers ad let m ad be ratioal umbers. The followig properties have the same ames as those listed o page 0, but ow appl to ratioal epoets as illustrated. Propert Eample 1. a m p a a m 1 1/ p / (1/ 1 /). (a m ) a m ( / ) (/ p ). (ab) m a m b m (16 p 9) 1/ 16 1/ p 9 1/ p 1. a m 1 a m, a Þ 0 61/ 1 6 1/ 1 6. a m a am, a Þ 0 / 1/ (/ 1/) a b m am b m, b Þ / 71/ 6 1/ E XAMPLE 1 Use properties of epoets Use the properties of ratioal epoets to simplif the epressio. 7 1/ p 7 1/ 7 (1/ 1 1/) 7 / b. (6 1/ p 1/ ) (6 1/ ) p ( 1/ ) 6 (1/ p ) p (1/ p ) 6 1 p / 6 p / c. ( p ) 1/ [( p ) ] 1/ (1 ) 1/ 1 [ p (1/)] d. 1 1/ 1/ (1 1/) / e. 1 1/ 6 1/ F1 6 1/ G (7 1/ ) 7 (1/ p ) 7 / 0 Chapter 6 Ratioal Epoets ad Radical Fuctios

2 E XAMPLE Appl properties of epoets BIOLOGY A mammal s surface area S (i square cetimeters) ca be approimated b the model S km / where m is the mass (i grams) of the mammal ad k is a costat. The values of k for some mammals are show below. Approimate the surface area of a rabbit that has a mass of. kilograms (. 10 grams). Mammal Sheep Rabbit Horse Huma Moke Bat k Solutio S km / Write model. 9.7(. 10 ) / Substitute 9.7 for k ad. 10 for m. 9.7(.) / (10 ) / Power of a product propert ø 9.7(.6)(10 ) Power of a power propert ø 00 Simplif. c The rabbit s surface area is about 00 square cetimeters. GUIDED PRACTICE for Eamples 1 ad Simplif the epressio. 1. ( 1/ p 7 1/ ). / p 1/. 1/. 1 01/ 1/. BIOLOGY Use the iformatio i Eample to approimate the surface area of a sheep that has a mass of 9 kilograms (9. 10 grams). PROPERTIES OF RADICALS The third ad sith properties o page 0 ca be epressed usig radical otatio whe m 1 for some iteger greater tha 1. KEY CONCEPT For Your Notebook Properties of Radicals Product propert of radicals Ï a p b Ï a p Ï b Î a b Quotiet propert of radicals Ï a Ï, b? 0 b E XAMPLE Use properties of radicals Use the properties of radicals to simplif the epressio. b. Ï 1 p Ï 18 Ï 1 p 18 Ï 16 6 Product propert Ï 80 Ï Î 80 Ï 16 Quotiet propert 6. Appl Properties of Ratioal Epoets 1

3 SIMPLEST FORM A radical with ide is i simplest form if the radicad has o perfect th powers as factors ad a deomiator has bee ratioalized. E XAMPLE Write radicals i simplest form Write the epressio i simplest form. Ï 1 Ï 7 p Factor out perfect cube. Ï 7 p Ï Product propert Ï Simplif. REVIEW RADICALS For help with ratioalizig deomiators of radical epressios, see p. 66. b. Ï 7 Ï 8 Ï 7 Ï 8 p Ï Ï Ï 8 Ï Make deomiator a perfect fifth power. Product propert Ï 8 Simplif. LIKE RADICALS Radical epressios with the same ide ad radicad are like radicals. To add or subtract like radicals, use the distributive propert. E XAMPLE Add ad subtract like radicals ad roots Simplif the epressio. Ï Ï 10 (1 1 7) Ï 10 8 Ï 10 b. (8 1/ ) 1 10(8 1/ ) ( 1 10)(8 1/ ) 1(8 1/ ) c. Ï Ï Ï 7 p Ï Ï Ï Ï ( 1) Ï Ï GUIDED PRACTICE for Eamples,, ad Simplif the epressio. 6. Ï 7 p Ï 7. Ï 0 Ï 8. Î 9. Ï 1 Ï 0 VARIABLE EXPRESSIONS The properties of ratioal epoets ad radicals ca also be applied to epressios ivolvig variables. Because a variable ca be positive, egative, or zero, sometimes absolute value is eeded whe simplifig a variable epressio. Whe is odd Whe is eve Rule Ï Ï Eample 7 Ï 7 ad Ï 7 () 7 Ï ad Ï () Absolute value is ot eeded whe all variables are assumed to be positive. Chapter 6 Ratioal Epoets ad Radical Fuctios

4 E XAMPLE 6 Simplif epressios ivolvig variables Simplif the epressio. Assume all variables are positive. Ï 6 6 Ï ( ) Ï p Ï ( ) b. (7p q 1 ) 1/ 7 1/ (p ) 1/ (q 1 ) 1/ p ( p 1/) q (1 p 1/) pq c. Î m 8 Ï m Ï 8 Ï m Ï ( ) m d. 1 1/ / z 6 7(1 /) 1/ z (6) 7 1/ 1/ z 6 E XAMPLE 7 Write variable epressios i simplest form Write the epressio i simplest form. Assume all variables are positive. Ï a 8 b 1 c Ï a a b 10 b c Factor out perfect fifth powers. Ï a b 10 c p Ï a b Product propert AVOID ERRORS You must multipl both the umerator ad deomiator of the fractio b so that the value of the fractio does ot chage. b. Î 8 Î ab c Ï a b Simplif. p 8 p Î 9 Ï Ï 9 Make deomiator a perfect cube. Simplif. Quotiet propert Ï Simplif. E XAMPLE 8 Add ad subtract epressios ivolvig variables Perform the idicated operatio. Assume all variables are positive. 1 Ï w 1 Ï w Ï w Ï w b. 1/ 8 1/ ( 8) 1/ 1/ c. 1 Ï z z Ï z 1z Ï z z Ï z (1z z) Ï z 9z Ï z GUIDED PRACTICE for Eamples 6, 7, ad 8 Simplif the epressio. Assume all variables are positive. 10. Ï 7q Î / 1/ 1/ 1. Ï 9w wï w 6. Appl Properties of Ratioal Epoets

5 6. EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS o p. WS1 for Es., 7, ad 8 TAKS PRACTICE AND REASONING Es., 1, 69, 86, 89, 91, ad 9 1. VOCABULARY Are Ï ad Ï like radicals? Eplai wh or wh ot.. WRITING Uder what coditios is a radical epressio i simplest form? EXAMPLE 1 o p. 0 for Es. 1 PROPERTIES OF RATIONAL EXPONENTS Simplif the epressio.. / p 1/. (6 / ) 1/. 1/ p 7 1/ / / 1/ / / 11 / 10. (1 / p 8 / ) / p 10 / 7 / 1. 6 /9 p 6 /9 / 1. (16 /9 p 7/9 ) 1. 1 /7 1 /7 EXAMPLE o p. 1 for Es. 1 PROPERTIES OF RADICALS Simplif the epressio. 1. Ï 0 p Ï Ï 6 Ï 0. Ï 16 p Ï 17. Ï Ï 7 1. Ï 8 p Ï ( Ï p Ï ) 1 Ï 6 p Ï 9 Ï. Ï 8 p Ï 16 8 Ï p 8 Ï EXAMPLE o p. for Es. 1. MULTIPLE TAKS REASONING CHOICE What is the simplest form of the epressio Ï p (6Ï )? A Ï 10 B 18 Ï 10 C 6 Ï 10 D 6 Ï 8 10 SIMPLEST FORM Write the epressio i simplest form.. Ï 7. 6 Ï 6 6. Ï 108 p Ï 7. Ï 6 p Ï 8 8. Î Ï Î Ï 9 Ï 7 EXAMPLE o p. for Es. 1 COMBINING RADICALS AND ROOTS Simplif the epressio.. 6 Ï Ï. Ï 1 Ï. Ï 1 Ï. 1 8 Ï Ï Ï 1 Ï Ï 1 7 Ï Ï 7 Ï 1 9. Ï 10 8 Ï 0. Ï 8 Ï 70 ERROR ANALYSIS Describe ad correct the error i simplifig the epressio. 1. Ï Ï ( 1 6) Ï 1. Î Î p Î 8 Ï 1 Ï Chapter 6 Ratioal Epoets ad Radical Fuctios

6 EXAMPLE 6 o p. for Es. 1 VARIABLE EXPRESSIONS Simplif the epressio. Assume all variables are positive.. 1/ p 1/. ( ) 1/6. Ï / 7. / 1/ 8. Î Ï p 6 Ï 0. Ï p Ï Ï OPEN-ENDED TAKS REASONING MATH Write two variable epressios with oiteger epoets whose quotiet is /. EXAMPLE 7 o p. for Es. 9 EXAMPLE 8 o p. for Es SIMPLEST FORM Write the epressio i simplest form. Assume all variables are positive.. Ï Ï 7. Î 6 Ï 1 6 z 1. Ï p Ï 1. Ï z p Ï z 8. Î 0 9z COMBINING VARIABLE EXPRESSIONS Perform the idicated operatio. Assume all variables are positive. 60. Ï 1 9 Ï Ï 6 7 Ï / 1 / 6. 7 Ï 1 16 Ï 6. ( ) 1/ 1 ( 1/ ) 6. Ï 9 Ï 6. Ï 6 1 Ï 16 GEOMETRY Fid simplified epressios for the perimeter ad area of the give figure. 66. / 67. 1/ 68. 1/ 7 1/ 1/ 69. MULTIPLE TAKS REASONING CHOICE What is the simplified form of 1 Ï 1 Ï 9? 6 6 A 1 Ï B 1 Ï 6 C 6 Ï D 6 Ï 6 DECIMAL EXPONENTS Simplif the epressio. Assume all variables are positive p p 6 7. ( 6 ) ( ) ( 0. ) 77. 1z 0. (z 0.1 ) IRRATIONAL EXPONENTS The properties i this lesso ca also be applied to irratioal epoets. Simplif the epressio. Assume all variables are positive. 78. Ï Ï 79. ( Ï ) Ï p p/ 81. Ï 1 Ï 8. CHALLENGE Solve the equatio usig the properties of ratioal epoets. 9 b. p c. ( ) Appl Properties of Ratioal Epoets

7 PROBLEM SOLVING EXAMPLE o p. 1 for Es BIOLOGY Look back at Eample o page 1. Use the model S km / to approimate the surface area of the mammal give its mass. Bat: grams b. Huma: 9 kilograms 8. AIRPLANE VELOCITY The velocit v (i feet per secod) of a jet ca be approimated b the model v 8.8Î L A where A is the area of the wigs (i square feet) ad L is the lift (i Newtos). Fid the velocit of a jet with a wig area of. 10 square feet ad a lift of Newtos. 8. PINHOLE CAMERA The optimum diameter d (i millimeters) of the pihole i a pihole camera ca be modeled b d 1.9F(. 10 )lg 1/ where l is the legth of the camera bo (i millimeters). Fid the optimum pihole diameter for a camera bo with a legth of 10 cetimeters. 86. SHORT TAKS REASONING RESPONSE Show that the hpoteuse of a isosceles right triagle with legs of legth is Ï. 87. STAR MAGNITUDE The apparet magitude of a star is a umber that idicates how fait the star is i relatio to other stars. The epressio.1 m 1.1 m tells how ma times faiter a star with magitude m 1 is tha a star with magitude m. How ma times faiter is Altair tha Vega? b. How ma times faiter is Deeb tha Altair? c. How ma times faiter is Deeb tha Vega? Star Apparet magitude Costellatio Vega 0.0 Lra Altair 0.77 Aquila Deeb 1. Cgus 88. PHYSICAL SCIENCE The maimum horizotal distace d that a object ca travel whe lauched at a optimum agle of projectio is give b d v 0Ï (v0 ) 1 gh 0 g where h 0 is the object s iitial height, v 0 is its iitial speed, ad g is the acceleratio due to gravit. Simplif the model whe h 0 0. WORKED-OUT SOLUTIONS 6 Chapter 6 Ratioal p. WS1 Epoets ad Radical Fuctios TAKS PRACTICE AND REASONING

8 89. EXTENDED TAKS REASONING RESPONSE You have filled two roud balloos with water. Oe balloo cotais twice as much water as the other balloo. Solve the formula for the volume of a sphere, V πr, for r. b. Substitute the epressio for r from part (a) ito the formula for the surface area of a sphere, S πr. Simplif to show that S (π) 1/ (V ) /. c. Compare the surface areas of the two water balloos usig the formula from part (b). 90. CHALLENGE Substitute differet combiatios of odd ad eve positive itegers for m ad i the epressio Ï m. If is ot alwas positive, whe is absolute value eeded i simplifig the epressio? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzoe.com REVIEW Lesso.; TAKS Workbook 91. TAKS PRACTICE Which equatio best represets a lie parallel to the lie show? TAKS Obj. 7 A B 1 C 1 D (, ) 1 (, ) REVIEW Lesso 1.6; TAKS Workbook 9. TAKS PRACTICE What is the solutio of the iequalit 6 1 1? TAKS Obj. F 1 G H 1 J QUIZ for Lessos Evaluate the epressio without usig a calculator. (p. 1) 1. 6 /. 6 /. (6 / ). () / Solve the equatio. Roud our aswer to two decimal places whe appropriate. (p. 1) ( 1 ) 16 Simplif the epressio. Assume all variables are positive. (p. 0) 9. Ï p Ï (Ï 10 p Ï 10) ( 6 ) 1/8 1 ( 1/ 1/ ) 1. Ï 7 1 Ï 7 Ï 7 1. Ï p Ï Ï Ï Ï GEOMETRY Fid a radical epressio for the perimeter of the red triagle iscribed i the square show to the right. Simplif the epressio. (p. 0) 8 EXTRA PRACTICE for Lesso 6., p ONLINE Appl Properties QUIZof at Ratioal classzoe.com Epoets 7