11.1 Radical Expressions and Rational Exponents

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1 1 m Locker LESSON 11.1 Radical Expressios ad Ratioal Expoets Name Class Date 11.1 Radical Expressios ad Ratioal Expoets Essetial Questio: How are ratioal expoets related to radicals ad roots? Resource Locker Commo Core Math Stadards The studet is expected to: NRN.1 Explai how the defiitio of the meaig of ratioal expoets follows from extedig the properties of iteger expoets to those values, allowig for a otatio for radicals i terms of ratioal expoets. Mathematical Practices MP.6 Precisio Laguage Objective Idetify, with a parter, matchig radical expressios ad ratioal equatios. 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 ) = ( a m ) Defiitio of square root a = ( a m ) ENGAGE Essetial Questio: How are ratioal expoets related to radicals ad roots? Possible aswer: Ratioal expoets ad radicals are both ways to represet roots of quatities. The deomiator of a ratioal expoet ad the idex of a radical represet the root. The ratioal expoet m o a quatity represets the mth power of the th root of the quatity or the th root of the mth power of the quatity, where is the idex of the radical. Houghto Miffli Harcourt Publishig Compay Power of a power property a = a Defiitio of first power a = a m The bases are the same, 1 = m 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 a = ( a m ) 1 m PREVIEW: LESSON PERFORMANCE TASK View the Egage sectio olie. Discuss the photo ad how both the air temperature ad the wid speed ca cotribute to the wid chill. The preview the Lesso Performace Task. Power of a power property a = a m Module 11 Lesso 1 Name Class Date 11.1 Radical Expressios ad Ratioal Expoets Essetial Questio: How are ratioal expoets related to radicals ad roots? NRN.1 For the full text of this stadard, see the table startig o page CA. Houghto Miffli Harcourt Publishig Compay 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. 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 ) = ( a m ) Defiitio of square root = ( a m ) Power of a power property a = a Defiitio of first power a = a m The bases are the same, = so equate expoets. Solve. m = So, a = a. How ca you express a cube root usig a expoet? That is, if Give a = a m Cube both sides. ( a ) = ( a m ) Defiitio of cube root = Power of a power property = a 1 a m a (a m ) a m a = a m, what is m? Resource Module 11 Lesso 1 HARDCOVER PAGES 1 Tur to these pages to fid this lesso i the hardcover studet editio. Lesso 11.1

2 Defiitio of first power = The bases are the same, 1 = m so equate expoets. Solve. m = So, a 1 a m a = a. EXPLORE Defiig Ratioal Expoets i Terms of Roots 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? Yes; the oly differece is that istead of squarig or cubig both sides ad usig the defiitio of square root or cube root, you raise both sides to the th power ad use the defiitio of th root. The other reasoig is exactly the same. You ca coclude that fidig the power is the same as fidig the th root, or that a = a.. For a positive umber a, uder what coditio o will there be oly oe real th root? two real th roots? Explai. Whe is odd; whe is eve; for a odd power like x or x, every distict value of x gives a uique result, so there is oly oe umber that raised to the power will give the result, or oe th root. For example, there is oe fifth root of because is the oly umber whose fifth power is. For a eve power like x or x, there are two values of x (opposites of each other) that raised to the power will give the result. For example, there are two fourth roots of 1 because ad () both equal 1.. For a egative umber a, uder what coditio o will there be o real th roots? oe real th root? Explai. Whe is eve; whe is odd; o eve power of ay umber is egative, but every umber positive or egative has exactly oe th root whe is odd. Module 11 Lesso 1 PROFESSIONAL DEVELOPMENT Learig Progressios I this lesso, studets lear about ratioal expoets, ad how to traslate betwee radical expressios ad expressios cotaiig ratioal expoets. Studets will use these skills i the ext lesso, where they will lear how to apply the properties of ratioal expoets to simplify expressios cotaiig radicals or ratioal expoets. They will also apply these skills to the solvig of realworld problems that ca be modeled by radical fuctios. Houghto Miffli Harcourt Publishig Compay INTEGRATE TECHNOLOGY Studets have the optio of completig the Explore activity either i the book or olie. QUESTIONING STRATEGIES Whe rewritig a radical expressio by usig a ratioal expoet, where do you place the idex of the radical? i the deomiator of the ratioal expoet How does kowig that a = a help you to simplify the expressio 16 0.? You ca rewrite the fractio as. So = 16 = 16 =. AVOID COMMON ERRORS Studets may eed to be remided that although, for example, both ad are fourth roots of 1, the expressio 1 idicates the positive (pricipal) fourth root of 1, or. Thus, the expressio 1 simplifies to, ot to both ad. CONNECT VOCABULARY I order to use accurate laguage i their explaatios ad questios, studets eed to uderstad the differece betwee the terms radical ad radicad. Some studets may ot be familiar with the term radicad. Explai that it is the umber or expressio uder the radical sig. Whe covertig a radical expressio to a power, the radicad becomes the base of the power. Radical Expressios ad Ratioal Expoets

3 EXPLAIN 1 Traslatig Betwee Radical Expressios ad Ratioal Expoets QUESTIONING STRATEGIES What does the umerator of a ratioal expoet idicate? the power of the expressio What does the deomiator idicate? the idex of the radical Explai 1 Traslatig Betwee Radical Expressios ad Ratioal Expoets I the Explore, you foud that a ratioal expoet m 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. The expoet m idicates the th root of the mth power of a quatity. 7 = ( m 7 ) = = 9a = ( a m ) = = m 6 = a = a m INTEGRATE MATHEMATICAL PRACTICES Focus o Critical Thikig MP. Discuss with studets that whe fidig the value of a umber raised to a ratioal expoet, they ca evaluate either the root or the power first. Lead them to see that it is ofte easier to evaluate the root first, sice doig so eables studets to work with smaller umbers. Houghto Miffli Harcourt Publishig Compay Notice that you ca evaluate each example i the Numbers colum usig the equivalet defiitio. 7 = 7 = 79 = 9 = ( ) = = Example 1 Traslate radical expressios ito expressios with ratioal expoets, ad vice versa. Simplify umerical expressios whe possible. Assume all variables are positive. a. (1) b. x 11 c. 6 d. x a. (1) = ( 1) = () = 6 b. x 11/ = c. 6 = 6 d. x 11 or ( x ) 11 x = x a. 1 ( 16) a. ( 16) 1 ( = 6 b. (xy) c. ) = ( ) d. ( x y ) = 7 b. (xy) = (xy) or ( xy ) c = 11 = 11 = 11 d. x ( y ) = x ( y ) Module 11 Lesso 1 COLLABORATIVE LEARNING PeertoPeer Activity Have studets work i pairs. Istruct each pair to create a quiz for aother pair to take. Have them create five questios that ivolve simplifyig a power that cotais a ratioal expoet, ad that cotai expressios which ca be simplified without the use of a calculator. Have pairs exchage quizzes with other pairs, work o the quiz with their parters, ad check their aswers with the pair that created the quiz. Lesso 11.1

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? As a power of a root: Eter to obtai 0.1. The eter 0.1 to obtai As a root of a power: Eter The fid (As.). The result is agai Your Tur. Traslate radical expressios ito expressios with ratioal expoets, ad vice versa. Simplify umerical expressios whe possible. Assume all variables are positive. a. ( ) b. (y) b c ( ) = ( ) = ( ) = 9 c d. ( u st ) v = 0. Explai Modelig with Power Fuctios The followig fuctios all ivolve a give power of a variable. (y) b c = ( c y ) b or c (y) b = 0. = 0.1 ( u st ) v = (st) v u EXPLAIN Modelig with Power Fuctios QUESTIONING STRATEGIES How is a power fuctio related to a radical fuctio? It is the same as the related radical fuctio, just a way of expressig the fuctio with a ratioal expoet istead of a radical. How do you idetify the restrictios o the domai of a power fuctio that represets a realworld situatio? The domai must be restricted to umbers that make x b a real umber, ad further restricted to umbers that make sese i the cotext of the situatio. A = π r (area of a circle) V = π r (volume of a sphere) T = 1.11 L (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 Solve each problem by modelig with power fuctios. 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. Typical Body Mass Aimal Mass (kg) House cat. Cheetah Lio 170 Houghto Miffli Harcourt Publishig Compay Image Credits: Radius Images/Corbis Module 11 6 Lesso 1 DIFFERENTIATE INSTRUCTION Cogitive Strategies Studets who cotiue to cofuse the coversio of the umerator ad deomiator of the ratioal expoet to the expoet ad idex of the related radical expressio may beefit from writig e (e for expoet, i for idex) ext to i the ratioal expoet before covertig to the radical expressio. Radical Expressios ad Ratioal Expoets 6

5 INTEGRATE TECHNOLOGY A graphig calculator ca be used to explore the graph of the power fuctio i the example. Studets ca also use the TABLE feature to idetify the value of the fuctio for differet values of the domai. 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 times that of the house cat. Is the lio s metabolic rate more or less tha 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) = (.). The metabolic rate for the lio is oly about 1 times that of the house cat. The fuctio h (m) = 1 m 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? Houghto Miffli Harcourt Publishig Compay 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 0.01 for m i the formula ad use a calculator. h (m) = 1 ( 0.01 ) 760 The model estimates the shrew s restig heart rate to be about 760 beats per miute. b. Substitute 00 for m i the formula ad use a calculator. h (m) = 1 ( 00 ) The model estimates the elk s restig heart rate to be about beats per miute. c. Fid the ratio of h(m) for the larger smaller aimal to the aimal. Let 1 represet the mass of the smaller aimal = 10 = % You would expect the larger aimal s restig heart rate to be about of the smaller aimal s restig heart rate. Module 11 7 Lesso 1 7 Lesso 11.1

6 Reflect 6. What is the differece betwee a power fuctio ad a expoetial fuctio? A power fuctio ivolves a give power of a variable, while a expoetial fuctio ivolves a variable power of a give umber (the base). 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. Yes; a power with a egative iteger expoet is the reciprocal of the correspodig positive power. So, for example, for the elk, this would mea that 00 = 1. Usig the 00 calculator agai, h (m) = 1 ( 60, which is cosistet. ) 00 Your Tur. 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. Kleiber s law for lio: 7. (170) 0 Calories Kleiber s law for huma: 7. (70) 170 Calories 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. If metabolic rate ad mass were directly proportioal the 0 Cal 170 kg = x Cal 70 kg, so 170x = (0)(70), or x = ( 1, ) 100 Cal. so, the rate for a huma would be sigificatly lower tha the actual predictio from Kleiber s law. Kleiber s law idicates that smaller orgaisms have a higher metabolic rate per kilogram of mass tha do larger orgaisms. Elaborate 9. Explai how ca you use a radical to write ad evaluate the power.. You ca first rewrite the decimal as the fractio. The. = 10. Whe y = kx for some costat k, y varies directly as x. Whe y = k x, 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? y varies directly as the threefifths power of x. 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. They are both true. For a real umber a ad itegers m ad with 0, a m = ( m a ) = a m, so the order i which you fid the root or power does ot matter. 10 = = ( ) = =. Houghto Miffli Harcourt Publishig Compay CONNECT VOCABULARY For Eglish laguage learers, differetiatig betwee the words ratioal ad radical ca be difficult, both i prit ad i speech. Cotiue to make explicit coectios betwee the terms meaigs ad symbols each time they are used. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus o Reasoig MP. Ask studets to cosider how they ca use the fact that a m = a m to prove that k k =, give that k is ay positive iteger. The ask them to create some examples usig other bases ad differet values of k to verify this idetity. SUMMARIZE THE LESSON How ca you rewrite a radical expressio as a expoetial expressio ad vice versa? You ca write a radical expressio as the radicad raised to a fractio i which the umerator is the power of the radicad ad the deomiator is the idex of the radical. You ca write a expoetial expressio with the base of the expoet as the radicad, the deomiator of the expoet as the idex, ad the umerator of the expoet as the power. Module 11 Lesso 1 LANGUAGE SUPPORT Visual Cues Have studets work i pairs. Provide each pair with idex cards o which are writte either ratioal expressios or matchig radical expressios. Iclude some radical expressios with square roots, ratioal expressios with ratioal expoets with a umerator of 1, ad so o. Have studets match cards ad use colors or shapes to circle matchig powers ad idices. Suggest they write a as a idex of a square root, ad a 1 as a power, to show a appropriate match for the special cases metioed above. Radical Expressios ad Ratioal Expoets

7 EVALUATE 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 6 = ( 6 ) = = 10 p x q = x p q p or ( x ) Olie Homework Hits ad Help Extra Practice ASSIGNMENT GUIDE Cocepts ad Skills Explore Defiig Ratioal Expoets i Terms of Roots Example 1 Traslatig Betwee Radical Expressios ad Ratioal Expoets Example Modelig with Power Fuctios QUESTIONING STRATEGIES Practice Exercises 1 16 Exercises 17 0 If the radicad is a egative umber, what must be true about the idex? Explai. The idex must be odd. This is because eve roots of egative umbers are ot real umbers. (You ca t raise a real umber to a eve power ad get a egative umber.) VISUAL CUES Suggest that studets circle the deomiator i the ratioal expoet, ad draw a curved arrow from the deomiator, passig beeath the base, to a poit i frot of the expressio, idicatig that it becomes the idex of the radical i the coverted expressio. Houghto Miffli Harcourt Publishig Compay. (1). 7. ( ) ( 79 6 ) 6 = ( vw 0.6. () Traslate radical expressios ito expressios with ratioal expoets. Simplify umerical expressios whe possible. Assume all variables are positive (1) = ( 1 ) = () = = 7 = 9 vw = v 7 y y 7 = y = = = 7 6 (6) (πz) 6 (bcd) = (bcd) ( x ) ( x ) = ( 0.1 ) = 0. = 0.06 w or v ( w ) () 0.6 = () = ( ) = () = (bcd) = (bcd) = = = 79 6 ) = ( ) = 7 (6) 6 = (6) 6 7 (πz) = (πz) = 6 = 6 = 16 9 = ( 9 = ( x ) x ) = x = 6 x Module 11 9 Lesso 1 Exercise Depth of Kowledge (D.O.K.) Mathematical Practices Recall of Iformatio MP. Usig Tools 17 Skills/Cocepts MP. Reasoig 1 0 Skills/Cocepts MP. Modelig 1 1 Recall of Iformatio MP. Usig Tools Strategic Thikig MP. Reasoig Strategic Thikig MP. Logic 9 Lesso 11.1

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 1, 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? 6 ( 1 ) = 6 ( 6 1 ) = 6 ( ) = 6 ( ). The fret should be placed about. cm from the bridge. E strig Frets 6 cm Bridge AVOID COMMON ERRORS Whe usig a calculator to evaluate a expressio that cotais a ratioal expoet, studets ofte forget to put paretheses aroud the expoet. Use a example, such as, which studets ca simplify metally, to show that the value of the expressio whe etered without paretheses is ot the same as the value of the expressio whe etered correctly. 1. Meteorology The fuctio W = T.7 V + 0.7TV 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 F ad the wid speed is miles per hour. W = T.7 V + 0.7TV = ().7 () ()( ) 11.1 The widchill temperature is about 11 F. INTEGRATE TECHNOLOGY Studets ca use a graphig calculator to check their work. The MATH submeu, i the MATH meu, cotais a cube root fuctio as well as a fuctio that ca be used for radicals with other idices. 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 = 100 (T + 7) 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 = 6. Will the clump form a star? Justify your aswer. Yes; for this ad T, the Jea s mass is M J = 100 (1000) (6 + 7) = 100 (10) = The mass of the clump, 17, is greater tha the Jea s mass, 100, so the clump will form a star. Houghto Miffli Harcourt Publishig Compay Image Credits: (t) Zuzaa Dolezalova/Alamy; (b) Alex Tudorica/Shutterstock Module 11 0 Lesso 1 Radical Expressios ad Ratioal Expoets 0

9 INTEGRATE MATHEMATICAL PRACTICES Focus o Commuicatio MP. To help solidify studets uderstadig, have them verbalize their solutios to the exercises usig accurate laguage. For a problem ivolvig the simplificatio of, for example, a studet might describe the solutio i this way: Thirtytwo raised to the threefifths power is equal to the fifth root of thirtytwo raised to the third power, which is equal to two raised to the third power, which is equal to eight. PEERTOPEER DISCUSSION Ask studets to work with a parter to determie two expressios of the form, a m where m is ot a iteger, that are equal i value. Have studets share their examples with the class ad look for commoalities. Possible aswers: ad, 7 ad 1 0. 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 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? Fid the ratio of wages for the larger metropolita area to the smaller oe. 9 9 a,000,000 9 =,000, a 70,000 70,000 The total wages for the larger metropolita area will be about. times as great as the total wages for the smaller metropolita area. 1. Which statemet is true? A. I the expressio x, x is the radicad. B. I the expressio (16) x, is the idex. m C. The expressio 10 represets the mth root of the th power of 10. D. 0 = 0 (xy) = xy E. H.O.T. Focus o Higher Order Thikig. 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. JOURNAL Have studets write two differet represetatios, oe as a radical ad the other as a power, of the pricipal fourth root of the cube of 1. The have them describe how they would fid this value. Houghto Miffli Harcourt Publishig Compay 1 10 ; the egative expoet meas to take the reciprocal of the correspodig positive power. The correspodig positive power is, so the correct etry is m. Critical Thikig The graphs of three fuctios of the form y = ax 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. For graph B, m =, that is, y = ax. This is because the graph is that of a lie, for which the expoet o x is 1 ad the graph has a costat rate of chage (slope). For graph A, m >. This is because for a power greater tha 1, the average rate of chage of the graph icreases as x icreases, that is, the graph gets steeper. For graph C, m <. This is because for a power less tha 1, the average rate of chage of the graph decreases as x icreases, that is, the graph gets less steep. 6 y A C x B Module 11 1 Lesso 1 1 Lesso 11.1

10 . 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.) If is odd you ca fid a real umber odd root for every real umber, positive or egative. But if is eve (so m is odd sice the fractio is i lowest terms), the you are tryig to fid a eve root of a egative umber (i a m, a m is egative), which is ot possible. Lesso Performace Task + 0.7T V relates the wid The formula W = T.7 V 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 F to the earest degree whe the air temperature is 0 F, what is the wid speed to the earest mile per hour? a. Start by substitutig the values for air temperature ad wid speed. W = T.7V + 0.7T V W = (0).7 () (0)() Rewrite the formula i terms of roots ad powers. W = (0).7 ( ) (0) ( ) Evaluatig the formula gives 7.69, which ca be rouded to give a wid chill of about F. b. Begi by substitutig i the values you kow. W = T.7V + 0.7T V = (0).7 (V) (0)(V) The evaluate what you ca ad the rewrite the formula to solve for V. = (V) (V) = (V) 7.6 = 1.6(V) 7.6 = V = V (.016 ) = V 0 V A wid speed of 0 miles per hour makes a air temperature of 0 F feel like F. Houghto Miffli Harcourt Publishig Compay AVOID COMMON ERRORS Whe solvig the equatio for V, some studets may raise each term i the equatio to the power. Ask studets what they eed to do first. Isolate the variable V o oe side of the equatio. Metio that after the equatio is i the form V 7.6 =, each side of the equatio ca be 1.6 raised to the power. INTEGRATE MATHEMATICAL PRACTICES Focus o Critical Thikig MP. Have studets discuss whether the wid chill is iflueced more by the air temperature or by the wid speed. Have studets explai how the fractioal expoets affect the ifluece of the wid speed. Ask studets if the wid speed would have a larger or smaller effect if the expoets were itegers. Module 11 Lesso 1 EXTENSION ACTIVITY Have studets derive a equatio for the wid chill whe the temperature is i degrees Celsius ad the wid speed is i kilometers per hour. Have studets explai how the two equatios are differet. Ask studets how the uit coversio affects the expoets to which the variables are raised. The expoets remai the same. The uit coversio affects oly the coefficiets of the terms. Scorig Rubric poits: Studet correctly solves the problem ad explais his/her reasoig. 1 poit: Studet shows good uderstadig of the problem but does ot fully solve or explai his/her reasoig. 0 poits: Studet does ot demostrate uderstadig of the problem. Radical Expressios ad Ratioal Expoets