Understanding Rational Exponents and Radicals
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1 x Locker LESSON. Uderstadig Ratioal Expoets ad Radicals Name Class Date. Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? A..A simplify umerical radical expressios ivolvig square roots. Also A..B Resource Locker Texas Math Stadards The studet is expected to: A..A Simplify umerical radical expressios ivolvig square roots. Also A..B Mathematical Processes A..F Aalyze mathematical relatioships to coect ad commuicate mathematical ideas. Laguage Objective.C.,.C.,.I.,.I. Explai how radicals ad ratioal expoets are related. Explore Uderstadig Iteger Expoets Recall that powers like are evaluated by repeatig the base () as a factor a umber of times equal to the expoet (). So. What about a egative expoet, or a expoet of 0? You caot write a product with a egative umber of factors, but a patter emerges if you start from a positive expoet ad divide repeatedly by the base. Startig with powers of : 7 Dividig a power of by is equivalet to reducig the expoet by. Complete the patter: _ _ _ 0 _ - _ - 7 _ _ _ _ _ ENGAGE Essetial Questio: How are radicals ad ratioal expoets related? Radicals ad ratioal expoets ca be coverted back ad forth ito oe aother, showig that they are two differet forms of otatio for the same mathematical idea. PREVIEW: LESSON PERFORMANCE TASK View the Egage sectio olie. Discuss the photo, ad the fact that carbo-, a radioactive isotope of carbo, occurs i trace amouts, makig up about part per trillio of the carbo i the atmosphere. The preview the Lesso Performace Task. Houghto Miffli Harcourt Publishig Compay -, - Iteger expoets less tha ca be summarized as follows: Words Numbers Variables Ay o-zero umber raised to the power of 0 is ; 0 0 is udefied x for x 0 (.) Ay o-zero umber raised to a egative power is equal to divided - by the same umber raised to the opposite, positive power. Reflect - x for x 0, x ad iteger.. Discussio Why does there eed to be a exceptio i the secod for the case of x 0? For a egative expoet, usig x 0 would put a 0 i the deomiator ad divisio by zero is ot defied. Module 55 Lesso Name Class Date. Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? A..A simplify umerical radical expressios ivolvig square roots. Also A..B Explore Uderstadig Iteger Expoets Houghto Miffli Harcourt Publishig Compay Resource Recall that powers like are evaluated by repeatig the base () as a factor a umber of times equal to the expoet (). So. What about a egative expoet, or a expoet of 0? You caot write a product with a egative umber of factors, but a patter emerges if you start from a positive expoet ad divide repeatedly by the base. Startig with powers of : Dividig a power of by is equivalet to the expoet by. Complete the patter: 7 - _ 7 _ _ _ 0 _ - _ - _ _ _ _, - Iteger expoets less tha ca be summarized as follows: Words Numbers Variables Ay o-zero umber raised to the power of 0 is ; 0 0 is udefied 0 (.) Ay o-zero umber raised to a egative power is equal to divided - by the same umber raised to the opposite, positive power. Reflect reducig 0 x 0 for x 0. Discussio Why does there eed to be a exceptio i the secod for the case of x 0? x - for x 0, ad iteger. For a egative expoet, usig x 0 would put a 0 i the deomiator ad divisio by zero is ot defied. Module 55 Lesso HARDCOVER PAGES 8 Tur to these pages to fid this lesso i the hardcover studet editio. 55 Lesso.
2 Explore Explorig Ratioal Expoets A radical expressio is a expressio that cotais the radical symbol,. For a, is called the idex ad a is called the radicad. must be a iteger greater tha. a ca be ay real umber whe is odd, but must be o-egative whe is eve. Whe, the radical is a square root ad the idex is usually ot show. You ca write a radical expressio as a power. First, ote what happes whe you raise a power to a power. ( ) ( ) ( ) ( ) 6, so ( ). I fact, for all real umbers a ad all ratioal umbers m ad, ( a m ) a m. This is called the Power of a Power Property. A radical expressio ca be writte as a expoetial expressio: a a k. Fid the value for k whe. Start with the equatio. a a k Square both sides. ( a ) ( a k ) Defiitio of square root a ( a k ) Power of a power property a k a Reflect. What do you thik will be the rule for other values of the radical idex? Other radicals ca be writte as a. Equate expoets. Solve for k. k k EXPLORE Uderstadig Iteger Expoets INTEGRATE TECHNOLOGY Studets have the optio of completig the activity either i the book or olie. QUESTIONING STRATEGIES What patter ca you use to evaluate egative expoets? As the value of the expoet decreases by, the value of the power is divided by the base. Explai Simplifyig Numerical Expressios with th Roots For ay iteger >, the th root of a is a umber that, whe multiplied by itself times, is equal to a. x a x a The th root ca be writte as a radical with a idex of, or as a power with a expoet of. A expoet i the form of a fractio is a ratioal expoet. a a The expressios are iterchageable, ad to evaluate the th root, it is ecessary to fid the umber, x, that satisfies the equatio x a. Example 6 Fid the root ad simplify the expressio. Covert to radical. 6 6 Rewrite radicad as a power. Defiitio of th root 8 + Covert to radicals. 8 + Rewrite radicads as powers Apply defiitio of th root. + Simplify. 6 Houghto Miffli Harcourt Publishig Compay How ca you evaluate a umber writte with a egative expoet? A umber with a egative expoet ca be writte as the reciprocal of the umber writte with a positive expoet. EXPLORE Explorig Ratioal Expoets QUESTIONING STRATEGIES Whe you covert betwee radical form ad ratioal expoet form, what are the restrictios o the radicad ad idex? Coversios are doe for all real umbers for which the radical is defied. The idex must be a positive iteger ad the power of the radicad must be a iteger. Module 56 Lesso PROFESSIONAL DEVELOPMENT Learig Progressios I this lesso, studets exted their kowledge of expoets to the properties of iteger ad ratioal expoets while allowig for a otatio for radicals i terms of ratioal expoets. Some key uderstadigs for studets are as follows: The defiitio b _ b, where b > ad is a positive iteger, is used to simplify expressios with ratioal expoets. The square root ad cube root of a umber ca be writte with ratioal expoets. EXPLAIN Simplifyig Numerical Expressios with th Roots QUESTIONING STRATEGIES Whe you write ( 5 ) with a fractioal expoet, what is the deomiator of the fractioal expoet? Why? ; the square root idicates a power of _. Uderstadig Ratioal Expoets ad Radicals 56
3 INTEGRATE MATHEMATICAL PROCESSES Focus o Modelig Review powers ad roots by reviewig ad 5 5 with studets. Have studets practice writig several similar examples. The preset the defiitio of b _ ad discuss the Example. Show studets two special _ cases: ad 0 _ 0 for all atural-umber values of. AVOID COMMON ERRORS With fractioal expoets with a umerator other tha, studets may cofuse the idex with the power. Write bas e expoet idex o the board for studets to use as a referece. Your Tur Explai Simplifyig Numerical Expressios with Ratioal Expoets Give that for a iteger greater tha, m_ b b, you ca use the Power of a Power Property to defie b for ay positive iteger m. b m_ b m m_ b b m ( b m ) Power of a Power Property ( b m ) ( b ) m Defiitio of b _ b m The defiitio of a umber raised to the power of m is the th root of the umber raised to the mth power. The power of m ad the th root ca be evaluated i either order to obtai the same aswer, although it is geerally easier to fid the th root first whe workig without a calculator. Example Simplify expressios with fractioal expoets EXPLAIN Simplifyig Numerical Expressios with Ratioal Expoets QUESTIONING STRATEGY How do you simplify 8 _? Determie the th root of 8, or. Whe you simplify the ratioal expoet, what does it mea if the simplified form is a iteger? If it is a fractio? If the expoet is a iteger, the fial form will ot cotai a radical sig. If the expoet is a fractio, the fial form will cotai a radical sig. Houghto Miffli Harcourt Publishig Compay 7 _ Defiitio of b m 7 ( Rewrite radicad as a power. ( 5 _ Defiitio of cube root 7 ) ) Defiitio of b m 5 _ ( 5 ) Rewrite radicad as a power. ( 5 ) Defiitio of square root 5 Module 57 Lesso 5 COLLABORATIVE LEARNING Peer-to-Peer Activity Have studets work i pairs. Studets take turs rollig both a red (r) ad a blue (b) umber cube. After each roll, the studet uses the umbers show o the cubes to complete the expressio 6 r b. The the studet simplifies the expressio or states that it caot be simplified. The other studet checks the aswer ad the rolls the umber cubes to decide the ext expressio. 57 Lesso.
4 YourTur 5. _ _ - 5 ( 5 ) ( 5 5 ) 8 Elaborate 5_ - _ ( ) 5 - ( ) ( ) 5 - ( Why ca you evaluate a odd root for ay radicad, but eve roots require o-egative radicads? Multiplyig a umber by a egative umber chages the sig, so that i a product with multiple factors, a odd umber of egative factors results i a egative product, while a eve umber of egative factors results i a positive product. Positive factors do ot chage the sig of a product. There is o way to make a egative product with a eve umber of idetical factors. For odd roots, a egative umber simply has a egative root sice a odd umber of egative factors results i a egative product, while a positive umber has a positive root. 8. I evaluatig powers with ratioal expoets with values like, why is it usually better to fid the root before the power? Would it chage the aswer to switch the order? The th root is a smaller umber tha the base, while evaluatig the power of m first requires fidig the th root of a larger umber tha the base. Roots of large umbers ca be foud by guessig, but smaller umbers are more familiar (you are more likely to simply recogize the root or pick it o the first guess) ad eve if a few guesses are required, it is easier to check with small umbers. No, switchig the order would ot chage the aswer.. Essetial Questio Check-I How ca radicals ad ratioal expoets be used to simplify expressios ivolvig oe or the other? Radical expressios are iterchageable with expoets of the form _. Powers with ratioal expoets ca be evaluated by covertig them ito radical expressios with idex. Radical expressios with powers ca sometimes be simplified by switchig to ratioal expoets ad usig the properties of powers. ) Houghto Miffli Harcourt Publishig Compay INTEGRATE TECHNOLOGY Ecourage the use of graphig calculators to check the results of simplifyig umerical radical expressios ad umerical expressios with ratioal expoets. Ask studets to use the followig sample problems to practice eterig expressios correctly ito their calculators: Eter 6 as 6^(/); eter as ^(/); eter - _ 5 as 5^(-/). Make sure studets uderstad the importace of icludig paretheses due to the order of operatios. ELABORATE QUESTIONING STRATEGIES Whe simplifyig a fractioal expoet with the form m, will you get a differet aswer if you fid the root first ad the raise the aswer to the power, or raise to a power first ad the take the root? Explai. No, the order does t matter. You get the same aswer either way, although it is ofte easier to take the root first. SUMMARIZE THE LESSON How do you simplify a equatio with a ratioal expoet? If the expoet has the form _, fid the th root m of the base. If the expoet has the form, fid the th root of the base raised to the mth power. Module 58 Lesso DIFFERENTIATED INSTRUCTION Kiesthetic Experiece As studets work o a problem, suggest that kiesthetic learers write the base, idex, ad power o separate small pieces of paper. Have studets arrage the pieces of paper to form the origial expressio. The have studets draw a radical o a sheet of paper ad move pieces of paper ito their correct positios i the radical. Uderstadig Ratioal Expoets ad Radicals 58
5 EVALUATE Evaluate: Homework ad Practice Evaluate the expressios _ _ 56 Olie Homework Hits ad Help Extra Practice ASSIGNMENT GUIDE Cocepts ad Skills Explore Uderstadig Iteger Expoets Explore Explorig Ratioal Expoets Example Simplifyig Numerical Expressios with th Roots Example Simplifyig Numerical Expressios with Ratioal Expoets Practice Exercises 6, 0 Exercises 6 7 Exercises 7 0, 8 Exercises, 5, Houghto Miffli Harcourt Publishig Compay. -. ( - 5. (-) 6 ( - ) ) - ( _ ) Fid the root(s) ad simplify the expressio (_ ) ( _ ) 6 -. ( _ 6 ). ( 6 6 ) Module 5 Lesso LANGUAGE SUPPORT Coect Vocabulary Write the terms 5 ad 5 o the board. Poit out that the first expressio, 5, is kow as the square root or the secod root of 5. Explai that, i more complicated radical expressios, such as 5, the iside expressio, 5, is called the radicad, while the root (, i the upper left of the radical symbol) is called the idex. 5 meas the third root, or cube root, of to the fifth power. 5 Lesso.
6 Simplify the expressios with ratioal expoets.. _. 8 5 ( ) ( 7 ) 7 8 5_ ( 8 ) 5 ( ) 5 5 INTEGRATE MATHEMATICAL PROCESSES Focus o Commuicatio Circulate as studets solve the problems. Ivite studets to explai their reasoig as they begi a ew problem.. 7 _ + _. 5 _ + 6 _ 7 _ + _ ( 7 ) + ( ) ( ) + ( ) ( 5 ) + ( ( 5 ) + ( 6 ) ) Simplify the expressios _ 8. 8 _ _ - _ 8 _ _ 8 _ + 8 _ ( 8 ) + ( 8 ) ( ) ( ) Houghto Miffli Harcourt Publishig Compay Module 50 Lesso Exercise Depth of Kowledge (D.O.K.) Mathematical Processes Skills/Cocepts.E Create ad use represetatios 5 Skills/Cocepts.A Everyday life 6 Skills/Cocepts.G Explai ad justify argumets 7 8 Strategic Thikig.G Explai ad justify argumets Strategic Thikig.F Aalyze relatioships Uderstadig Ratioal Expoets ad Radicals 50
7 INTEGRATE MATHEMATICAL PROCESSES Focus o Math Coectios Explai that, whe writig a expressio with a ratioal expoet as a radical, the power ca also be placed uder the radical sig. For example, 6 ca be writte as ( 6 ) or 6. However, it is usually more coveiet to evaluate the root ad the evaluate the power. AVOID COMMON ERRORS Studets will sometimes multiply the base by the egative expoet. Have these studets re-read the defiitio of a egative expoet. Poit out that 0 - meas a umber less tha oe, ot a umber less tha zero. Houghto Miffli Harcourt Publishig Compay. 5 _ 7 _ 5 _ _ ( ) _. 6 _ + 6 _ ( ) ( _ ) ( ) ( ) ( ) ( ) ( 8 ) Geometry The volume of a cube is related to the area of a face by the formula V A _ What is the volume of a cube whose face has a area of 00 c m? v 00 _ ( 00 ) ( 0 ) cm _ 000 _ _ Module 5 Lesso 5 Lesso.
8 . Biology The approximate umber of Calories, C, that a aimal eeds each day is give by C 7 m _, where m is the aimal s mass i kilograms. Fid the umber of Calories that a 6 kilogram dog eeds each day. C 7 (6) _ 7 ( 6 ) 7 ( ) QUESTIONING STRATEGIES What geeralizatio ca you make whe the radicad s expoet ad the idex are equal, as i? Whe the expoet i the radicad ad the idex are the same, the expressio simplifies to the base of the radicad. Thus, simplifies to Calories 5. Rocket Sciece Escape velocity is a measure of how fast a object must be movig to escape the gravitatioal pull of a plaet or moo with o further thrust. The escape velocity for the moo is give approximately by the equatio V 5600 _ ( d 000 ) _, where v is the escape velocity i miles per hour ad d is the distace from the ceter of the moo (i miles). If a luar lader thrusts upwards util it reaches a distace of 6,000 miles from the ceter of the moo, about how fast must it be goig to escape the moo s gravity? v 5600 (_ 6, ) It eeds to be movig at about 00 miles per hour. 6. Multiple Respose Which of the followig expressios caot be evaluated? a. b. (-) c. d. (-) e. 0 f _ b, e ad f are udefied. Houghto Miffli Harcourt Publishig Compay WilleeCole/Alamy INTEGRATE MATHEMATICAL PROCESSES Focus o Modelig Make sure studets uderstad that the deomiator i the expoet determies the idex i the radical equivalet. For example, i 6 _, the i the expoet idicates the th root of 6. Module 5 Lesso Uderstadig Ratioal Expoets ad Radicals 5
9 COLLABORATIVE LEARNING Have studets work i groups of three or four. To help studets master the cocepts i the lesso, have members of a small group create problems ivolvig ratioal expoets or radicals, the have them solve each other s problems. Ecourage studets to make their problems as elaborate as they like, but they must be able to supply the correct aswer for ay problem that they submit. JOURNAL I their jourals, have studets write the steps they would use to simplify H.O.T. Focus o Higher Order Thikig _ 7. Explai the Error Yua is asked to evaluate the expressio (-8) o his exam, ad writes that you caot evaluate a egative umber to a eve fractioal power. Is he correct, ad if so, why? If he is ot correct, what is the correct aswer? No, he is ot correct. It is oly eve umbers i the deomiator of a expoet that caot be evaluated with a egative base. With a odd deomiator ad a eve umerator i the expoet, there is o problem. The correct aswer is: (-8) _ ( -8) ( (-) (-) ) 8. Commuicate Mathematical Ideas Show that the th root of a umber, a, ca be expressed with a expoet of for ay positive iteger,. a a k Raise both sides to the th power. Defiitio of th root a ( a k ) Power of a Power Property Equate Expoets. k Solve for k. k a a k ( a ) ( a k ) Houghto Miffli Harcourt Publishig Compay. Explai the Error Gretche thiks she has figured out how to evaluate the square root of a egative umber. Explai why her solutio is flawed. (-) (-) (-) 0 (-) The she solves for (-) which is the same thig as _ -. (-) (-) (-) (-) But the square root of - caot be, sice, ot -. Gretche's method has o validity because it is based o calculatios ivolvig the square root of a egative umber, which is ot defied. It is also riddled with errors. Module 5 Lesso 5 Lesso.
10 Lesso Performace Task Carbo- datig is used to determie the age of archeological artifacts of biological (plat or aimal) origi. Items that are dated usig carbo- iclude objects made from boe, wood, or plat fibers. This method works by measurig the fractio of carbo- remaiig i a object. The fractio of the origial carbo- remaiig ca be expressed by the fuctio, f ( t_ 5700), where t is the legth of time sice the orgaism died. a. Fill i the followig table to see what fractio of the origial carbo- still remais after the passage of time. t ,00 7,00 t_ 5700 Fractio of Carbo- Remaiig 0 b. The duratio of 5700 years is referred to as the half-life of carbo- because the amout of carbo- drops i half 5700 years after ay startig poit (ot just t 0 years). Verify this property by comparig the amout of remaiig carbo- after,00 years ad 7,00 years. 8 c. Write the correspodig expressio for the remaiig fractio of uraium-, which has a half-life of about 80,000 years. _ a. f ( ) _, f (, ) _, f ( 7, ) 8 b. From the table, f (7,00) 8 8 ad f (,00). Therefore, 5700 years after,00, half of the carbo- is remaiig. t_ c. f ( 80,000 ) Houghto Miffli Harcourt Publishig Compay Blaie Harrigto III/ Alamy CONNECT VOCABULARY Some studets may ot be familiar with the term half-life ( t _ used i this Lesso Performace Task. ) Explai that the half-life of ay substace is the time it takes for the amout of the substace to decrease to half of what it origially was. To illustrate, draw a log lie o the board, the draw successively shorter lies: half as log as the origial lie; the _ as log; the _ as log; ad so o. The lies represet the 8 shrikig by half at equal time itervals. INTEGRATE MATHEMATICAL PROCESSES Focus o Modelig Have studets begi with the umber 0 ad take half of it, the half of that result, the half of that, ad so o. Discuss with studets whe there might o loger be aythig to take half of. Studets should cotiue takig half util it is apparet to them that they will ever reach 0, although they will come close. Discuss how a graph of ordered pairs (umber of times halved, result) would look. Module 5 Lesso EXTENSION ACTIVITY Have studets make a table of values for uraium-, similar to the table they made i Part A of the Lesso Performace Task for carbo-. The have studets describe ay patters they see i the tables as they substitute values for t. Studets will fid that the values i the last two colums are exactly like the values i the last two colums of the table for carbo-. Scorig Rubric poits: Studet correctly solves the problem ad explais his/her reasoig. 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. Uderstadig Ratioal Expoets ad Radicals 5
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