11.1 Radical Expressions and Rational Exponents

Transcription

1 Name Class Date 11.1 Radical Expressios ad Ratioal Expoets Essetial Questio: How are ratioal expoets related to radicals ad roots? Resource Locker Explore Defiig Ratioal Expoets i Terms of Roots Remember that a umber a is a th root of a umber b if a = b. As you kow, a square root is idicated by ad a cube root by. I geeral, the th root of a real umber a is idicated by a, where is the idex of the radical ad a is the radicad. (Note that whe a umber has more tha oe real root, the radical sig idicates oly the pricipal, or positive, root.) A ratioal expoet is a expoet that ca be expressed as m, where m is a iteger ad is a atural umber. You ca use the defiitio of a root ad properties of equality ad expoets to explore how to express roots usig ratioal expoets. A How ca you express a square root usig a expoet? That is, if _ a = a m, what is m? Give _ a = a m Square both sides. ( _ a ) 2 = ( a m ) 2 Defiitio of square root = ( a m ) 2 Power of a power property a = a Defiitio of first power a = a 2m The bases are the same, = so equate expoets. Solve. m = So, _ a = a. B How ca you express a cube root usig a expoet? That is, if a = a m, what is m? Give a = a m Cube both sides. ( _ a ) = ( a m ) Defiitio of cube root = Power of a power property = Module Lesso 1

2 Defiitio of first power = The bases are the same, = so equate expoets. Solve. m = So, a = a. Reflect 1. Discussio Examie the reasoig i Steps A ad B. Ca you apply the same reasoig for ay th root, a, where is a atural umber? Explai. What ca you coclude? 2. For a positive umber a, uder what coditio o will there be oly oe real th root? two real th roots? Explai.. For a egative umber a, uder what coditio o will there be o real th roots? oe real th root? Explai. Module Lesso 1

3 Explai 1 Traslatig Betwee Radical Expressios ad Ratioal Expoets I the Explore, you foud that a ratioal expoet m 1 with m = 1 represets a th root, or that a = a for positive values of a. This is also true for egative values of a whe the idex is odd. Whe m 1, you ca thik of the umerator m as the power ad the deomiator as the root. The followig ways of expressig the expoet m are equivalet. Ratioal Expoets For ay atural umber, iteger m, ad real umber a whe the th root of a is real: Words Numbers Algebra The expoet m idicates the mth power of the th root of a quatity. 27 = ( 2 m_ 27 ) 2 = = 9a = ( a m ) The expoet m idicates the th root of the mth power of a quatity. _ 2 = = m_ 6 = 8a = a m Notice that you ca evaluate each example i the Numbers colum usig the equivalet defiitio. 27 = 27 2 = 729 = 9 _ 2 = ( _ ) = 2 = 8 Example 1 Traslate radical expressios ito expressios with ratioal expoets, ad vice versa. Simplify umerical expressios whe possible. Assume all variables are positive. A a. (12) _ b. x _ 11 8 c. 6 d. x _ a. (12) = ( 12) = () = 62 b. x 11/8 = c. 6 = 6 8 x 11 8 or ( x ) 11 _ d. x = x B a. _ 81 ( 16) _ a. _ 81 ( 16) = ( _ b. (xy) c. _ ) = ( ) _ b. (xy) = (xy) or ( xy ) 6 11 d. ( _ 2x y ) = c. d = 11 = 11 = _ 2x ( y ) = _ 2x ( y ) Module Lesso 1

4 Reflect. How ca you use a calculator to show that evaluatig as a power of a root ad as a root of a power are equivalet methods? Your Tur. Traslate radical expressios ito expressios with ratioal expoets, ad vice versa. Simplify umerical expressios whe possible. Assume all variables are positive. a. ( _ 2 b_ b. c 2) (y) c d. ( u st ) v Explai 2 Modelig with Power Fuctios The followig fuctios all ivolve a give power of a variable. A = π r 2 (area of a circle) V = π r (volume of a sphere) T = 1.11 L 1 2 (the time T i secods for a pedulum of legth L feet to complete oe backadforth swig) These are all examples of power fuctios. A power fuctio has the form y = ax b where a is a real umber ad b is a ratioal umber. Example 2 Solve each problem by modelig with power fuctios. A Biology The fuctio R = 7. M, kow as Kleiber s law, relates the basal metabolic rate R i Calories per day bured ad the body mass M of a mammal i kilograms. The table shows typical body masses for some members of the cat family. Aimal Typical Body Mass Mass (kg) House cat. Cheetah Lio 170 Image Credits: Radius Images/Corbis Module Lesso 1

5 a. Rewrite the formula with a ratioal expoet. b. What is the value of R for a cheetah to the earest 0 Calories? c. From the table, the mass of the lio is about 8 times that of the house cat. Is the lio s metabolic rate more or less tha 8 times the cat s rate? Explai. a. Because a m = a m, M = M, so the formula is R = 7. M. b. Substitute for M i the formula ad use a calculator. The cheetah s metabolic rate is about 100 Calories. c. Less; fid the ratio of R for the lio to R for the house cat. 7. (170) = 170 _ (.). The metabolic rate for the lio is oly about 1 times that of the house cat. B The fuctio h (m) = 21 m 1 models a aimal s approximate restig heart rate h i beats per miute give its mass m i kilograms. a. A commo shrew has a mass of oly about 0.01 kg. To the earest 10, what is the model s estimate for this shrew s restig heart rate? b. What is the model s estimate for the restig heart rate of a America elk with a mass of 00 kg? c. Two aimal species differ i mass by a multiple of 10. Accordig to the model, about what percet of the smaller aimal s restig heart rate would you expect the larger aimal s restig heart rate to be? a. Substitute for m i the formula ad use a calculator. 1_ h (m) = 21 ( ) The model estimates the shrew s restig heart rate to be about beats per miute. b. Substitute for m i the formula ad use a calculator. 1_ h (m) = 21 ( ) The model estimates the elk s restig heart rate to be about beats per miute. c. Fid the ratio of h(m) for the aimal to the aimal. Let 1 represet the mass of the smaller aimal. 1_ _ = = 1_ You would expect the larger aimal s restig heart rate to be about restig heart rate. of the smaller aimal s Module Lesso 1

6 Reflect 6. What is the differece betwee a power fuctio ad a expoetial fuctio? 7. I Part B, the expoet is egative. Are the results cosistet with the meaig of a egative expoet that you leared for itegers? Explai. Your Tur 8. Use Kleiber s law from Part A. a. Fid the basal metabolic rate for a 170 kilogram lio to the earest 0 Calories. The fid the formula s predictio for a 70 kilogram huma. b. Use your metabolic rate result for the lio to fid what the basal metabolic rate for a 70 kilogram huma would be if metabolic rate ad mass were directly proportioal. Compare the result to the result from Part a. Elaborate 9. Explai how ca you use a radical to write ad evaluate the power Whe y = kx for some costat k, y varies directly as x. Whe y = k x 2, y varies directly as the square of x; ad whe y = k x, y varies directly as the square root of x. How could you express the relatioship y = k x _ for a costat k? 11. Essetial Questio CheckI Which of the followig are true? Explai. To evaluate a expressio of the form a m_, first fid the th root of a. The raise the result to the mth power. To evaluate a expressio of the form a m_, first fid the mth power of a. The fid the th root of the result. Module Lesso 1

7 Evaluate: Homework ad Practice Traslate expressios with ratioal expoets ito radical expressios. Simplify umerical expressios whe possible. Assume all variables are positive x p q Olie Homework Hits ad Help Extra Practice. (12). 7 _. ( _ ) _ vw 8. (2) 0.6 Traslate radical expressios ito expressios with ratioal expoets. Simplify umerical expressios whe possible. Assume all variables are positive y (6) (bcd) 1. _ (πz) 9 ( _ x ) Module 11 8 Lesso 1

8 17. Music Frets are small metal bars positioed across the eck of a guitar so that the guitar ca produce the otes of a specific scale. To fid the distace a fret should be placed from the bridge, multiply the legth of the strig by 2 12, where is the umber of otes higher tha the strig s root ote. Where should a fret be placed to produce a F ote o a B strig (6 otes higher) give that the legth of the strig is 6 cm? E strig Frets 6 cm Bridge 18. Meteorology The fuctio W = T.7 V TV relates the widchill temperature W to the air temperature T i degrees Fahreheit ad the wid speed V i miles per hour. Use a calculator to fid the wid chill temperature to the earest degree whe the air temperature is 28 F ad the wid speed is miles per hour. 19. Astroomy New stars ca form iside a cloud of iterstellar gas whe a cloud fragmet, or clump, has a mass M greater tha the Jea s mass M J. The Jea s mass is M J = (T + 27) 2 where is the umber of gas molecules per cubic cetimeter ad T is the gas temperature i degrees Celsius. A gas clump has M =17, = 1000, ad T = 26. Will the clump form a star? Justify your aswer. Image Credits: (t) Zuzaa Dolezalova/Alamy; (b) Alex Tudorica/Shutterstock Module 11 8 Lesso 1

9 20. Urba geography The total wages W i a metropolita area compared to its total populatio p ca be approximated by a power fuctio of the form W = a p 9_ 8 where a is a costat. About how may times as great does the model predict the total earigs for a metropolita area with,000,000 people will be as compared to a metropolita area with a populatio of 70,000? 21. Which statemet is true? A. I the expressio 8 x, 8x is the radicad. B. I the expressio (16) x _, is the idex. m C. The expressio 102 represets the mth root of the th power of 102. D. 0 = 0 _ 2 (xy) = xy E. H.O.T. Focus o Higher Order Thikig 22. Explai the Error A teacher asked studets to evaluate 1 0 _ usig their graphig calculators. The calculator etries of several studets are show below. Which etry will give the icorrect result? Explai. 2. Critical Thikig The graphs of three fuctios of the form y = ax m are show for a specific value of a, where m ad are atural umbers. What ca you coclude about the relatioship of m ad for each graph? Explai. y 8 A 6 B C 2 x Module 11 8 Lesso 1

10 2. Critical Thikig For a egative real umber a, uder what coditio(s) o m ad ( 0) is a m a real umber? Explai. (Assume m is writte i simplest form.) Lesso Performace Task The formula W = T.7 V T V 2 relates the wid chill temperature W to the air temperature T i degrees Fahreheit ad the wid speed V i miles per hour. Fid the wid chill to the earest degree whe the air temperature is 0 F ad the wid speed is miles per hour. If the wid chill is about 2 F to the earest degree whe the air temperature is 0 F, what is the wid speed to the earest mile per hour? Module Lesso 1