Math 017. Materials With Exercises


 Lionel Powers
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1 Mth 07 Mterils With Eercises
2 Jul 0
3 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions Lesson Equivlent lgeric epressions... Lesson Opertions on power epressions with nonnegtive integer eponents Lesson Multipliction of lgeric epressions; The Distriutive Lw; Fctoriztion of common fctor; Fctoriztion of ; Simplifiction of lgeric frctions with the use of fctoriztion 0 Lesson 6 Addition nd sutrction of lgeric epressions.. 6 Lesson 7 Evlution of more complicted lgeric epressions; Sustitution of not onl numers ut lso lgeric epressions...70 Lesson 8 Generlities on equtions; Solving liner equtions in one unknown. 77 Lesson 9 Solving liner equtions involving frctions; Literl epressions: solving equtions for given vrile...88 Lesson 0 Liner Inequlities; Grphing sets of the tpe, on numer line..9 Lesson Recognizing nd mtching ptterns; Writing epressions in prescried w; Definition of liner eqution; Fctoriztion of the difference of squres Appendi A Solutions to ll eercises...6 Appendi B Smple tests..... Appendi C Smple tests solutions
4 Help ville: You cn find help in the Lerning L for mth in room B6 weekds nd in room B8 Mond Thursd evenings nd Sturds on Min Cmpus. During fll nd spring semesters, free, peer tutoring is ville eginning with the second week of clsses for ll current CCP students nd free, weekl workshops egin in the third week of the semester. The peer tutors re eperienced CCP students who hve tken mn of the courses in which the tutor. Mth specilists lso tutor s well s led workshops. Check t Regionl Centers for ds nd times of services. Also, during summer sessions, offerings m vr. Jul 0
5 Lesson Topics: Vriles nd lgeric epressions; Evlution of lgeric epressions. Vriles nd lgeric epressions s smolic representtions of numers Suppose tht ou thought of numer ut ou did not tell me wht it ws. I cn think out our numer s numer. The smol is n emple of vrile. Vrile A vrile is smol tht represents n unknown numer. The choice of the nme of vrile is ritrr. One cn s well cll it n, m or. We tret vriles s if the were numers. We cn, for emple, dd numers to vriles: m, or sutrct other vriles from them: m. We cn multipl them: m ; divide: ; rise to n m given power: m ; nd then, if we wish, dd ll of the epressions together: resulting epressions re clled lgeric epressions. m m. The m Algeric Epression An lgeric epression is numer, vrile or comintion of the two connected some mthemticl opertions like ddition, sutrction, multipliction, division, or eponentition. Notice tht numers nd vriles re lso emples of lgeric epressions. We cn refer to,, or s lgeric epressions. Just like,, or re numers (written in complicted w, ut numers), lgeric epressions m, or n re smolic representtions of numers. Both vriles nd lgeric epressions cn e thought of s unknown numers. Correct lnguge nd conventions used when forming lgeric epressions Algeric epressions re red using the sme terminolog s in rithmetic. For emple, A cn e red s A plus or the sum of A nd ; cn e red s rised to the second power or squred ; is red minus or the opposite of. The following convention is commonl dopted to indicte multipliction of numer nd vrile, or multipliction of vriles. To denote the opertion of multipliction, the sign of multipliction etween numer nd vrile or etween two vriles or epressions does not hve to e eplicitl displed, so for emple, A mens times A mens times, (+) mens times the quntit +
6 According to the ove convention the following is true. = = Although, it is customr to write insted of (just like n time we wnt to write, we just write not ). The following is lso true. = = Agin, it is customr to write insted of. When forming lgeric epressions, we plce prentheses ccording to the nottionl convention dopted in rithmetic. An time two opertion signs re net to ech other, prentheses re needed. For emple, we write: we do not write: ( ), ( ) ( ) Likewise, when multipling nd, we write ( ). The prentheses re needed even if the multipliction sign is not eplicitl displed. Notice tht if the prentheses re omitted, the epression chnges its mening from multipliction of nd to sutrction. Eponents pertin onl to the closest epression. You might, for emple, recll tht to indicte tht the entire frction is rised to given power, we use prentheses. We write. Similrl, whenever ou eponentite n lgeric epression tht is not represented single smol, ou must use prentheses:, do not require prentheses., ( ), Emple. How re the following epressions red? ) ) z c) m d) e) f) ( ). On the other hnd ) rised to the second power or squred ) times z, or the product of nd z, or multiplied z c) rised to the third power or cued d) rised to the mth power,
7 e) minus or the opposite of f) divided or the quotient of nd Emple. In the following epressions prentheses re needed. Eplin wh the re needed. n ) ( c) ) ( ) ) An time two opertion signs re net to ech other, prentheses re needed. In this cse the sign is followed sign. ) Without the prentheses, onl would e rised to the n th power. With prentheses, we rise to the nth power. The two sttements hve different mening, thus prentheses re needed. Algeric epressions llow us to epress mthemticl ides in generl form Algeric epressions llow us to write mthemticl ides in smols, without using specific numers. For emple, the re of squre is equl to the squre of the length of its side. Not ever squre is going to hve the sme size, so we use vrile to represent the length of side. If we denote s to e side of squre, then the re of the squre cn e epressed s s s s Emple. Let nd denote two different numers. Epress the following sttements using lgeric smols. ) The sum of nd ) The difference etween nd c) The product of nd d) The quotient of nd ) ) c) d) or equivlentl Emple. Find the lgeric epressions representing the opposite of the following epressions. Do not simplif. ) ) Recll tht to find the opposite of numer, we must multipl the numer, thus ) The opposite of is ) The opposite of is ( ) ( ). Plese, notice tht since the minus sign is directl followed nother minus sign, the prentheses re needed.
8 Emple. Use the letter to represent numer nd write the following sttements s lgeric epressions. ) Doule numer ) Two thirds of numer c) A quntit incresed ) ) c) Emple.6 Write the following sttements s lgeric epressions. ) sutrcted from A ) dded to A c) multiplied d) rised to the siteenth power ) A Notice the reversed order of vriles (if ou were sked to sutrct from 6, ou would write 6, similrl we write A ). ) ( A) Notice the use of prentheses: plus sign, followed minus sign, requires prentheses. c) ( )( ) or ( ) Notice the use of prentheses. A multipliction sign (even if not eplicitl displed), followed minus sign, requires prentheses. 6 d) Prentheses re needed to indicte tht the entire epression is rised to the siteenth power. Evlution of lgeric epressions As mentioned, we cn use vriles nd lgeric epressions to descrie certin quntittive reltionships without informtion out their specific vlues. For emple, suppose tht in certin store, cke costs dollrs. Let e vrile tht represents the numer of ckes we pln to u in tht store. To clculte how much we will p for such purchse, we multipl the price of one cke, dollrs, the numer of ckes we u,. Thus, we cn epress the cost s. The lgeric epression represents the cost of ckes ought t dollrs ech. From now on, n time we know the numer of ckes we wish to u, i.e. we know the vlue of, we cn find the price evluting the epression. To Evlute n Epression To evlute n lgeric epression mens to find its vlue once we know the vlues of its vriles. Ech vrile hs to e replced its vlue nd the resulting numericl epression hs to e clculted. 6
9 For emple, Evlute when 0 (in the ove emple it would men finding the price of 0 ckes t dollrs ech). 0 0 (The price of 0 ckes is 0 dollrs.) Notice, tht wht we did ws to sustitute 0 for in the epression. We could do tht ecuse is equl to 0. The following fundmentl principle underlies the process of evlution. If two quntities re equl, ou cn lws sustitute one for the other. Equls cn e sustituted for equls Cn n lgeric epression e evluted? Let us consider evlution of when 0. If we replce its vlue 0, the opertion we would hve to perform is division 0, ut division zero is not defined, so cnnot e evluted if 0. Epressions like, or 0 re undefined. 0 Emple.7 Rewrite ech epression replcing vriles with their vlues nd then evlute, if possile. If evlution is not possile, eplin wh it is not possile. ) 6, if ) C, if C c), if n d), if n e), if nd f), if Ech vrile should e replced its ssumed vlue nd the otined numericl sentence hs to e evluted. Plese, p ttention to the w prentheses re used. ) 6 6 ) C ( ) c) ( ) 6 Rememer tht mens multipliction of nd. n d) 8 e) ( ) f) The epression cnnot e evluted, since division 0 zero is undefined. 7
10 Common mistkes nd misconceptions Mistke. When, nd ou re sked to evlute, ou must e creful to write (). Do not forget to recop the minus sign efore. It is incorrect to evlute simpl writing. Mistke. One should NOT think tht lws represents negtive quntit. It depends on the vlue of. If, for emple,, then ( ). So we see tht if is negtive vlue, ctull represents positive quntit. Mistke. m When writing, do NOT plce m t the sme level s ut slightl higher. Otherwise, ecomes m. These two epressions hve different menings. Mistke. The epression is NOT red s two or two. Insted, red it s rised to the second power or squred. m Eercises with Answers (For nswers see Appendi A) Eercises 06 will help to review sic rithmetic opertions using integers, rtionl numers (frctions), decimls. E. Fill in lnks using the words: vrile(s), lgeric epression(s), nd numer(s) s pproprite. c,,, ( ) re emples of.,,,, c re emples of ut lso emples of. Vriles represent. If we know the vlue of, we cn evlute, nd s result we get. E. How re the following epressions red? ) ) c) e) f) cd g) m d) h) E. Rewrite the following epressions, inserting multipliction sign whenever multipliction is implied. Whenever there is no opertion of multipliction, clerl s so using the phrse there is no multipliction performed in this epression. ) 7 n ) km c) 8
11 d) ( ) e) f) z w( t) E. The opertion tht is indicted in the lgeric epression is, of course, ddition. Nme the opertion tht is to e performed in the following lgeric epressions. q ) ) c) s d) e) f) ( ) E. In the following epressions prentheses re needed. Eplin wh the re needed. ) ( ) m ) n c) ( c) d) ( ) e) ( ) f) ( ) 8 E.6 E.7 E.8 Determine which epression is rised to the nth power. n n ) ( s) ) s c) d) g) n st e) ) n ( st h) n f) n s i) n ( st) n ( s) Fill in the lnks. ) It is customr to write insted of. ) It is customr to write insted of. c) When one writes, it is understood tht the opertion tht is to e performed is. Fill in the lnks with numers to mke the sttement true. ) ) c) 0 n E.9 Write n lgeric epression representing the opposite numer of (do not remove prentheses). ) ) c) d) E.0 Use the letter to represent numer nd write the following phrses s lgeric epressions. ) Hlf of numer ) Three fourths of numer c) A quntit incresed 9
12 d) A numer sutrcted from v e) A quntit squred f) Three more thn numer g) A numer decresed h) The product of nd numer i) A numer douled E. Write the following phrses s lgeric epressions. Rememer to plce prentheses when needed (plce them onl when needed). Do not simplif. ) The sum of nd ) The difference of nd c) The product of nd d) The opposite of C f) The opposite of C g) The opposite of h) The product of v, t, nd p i) The quotient of c nd B j) rised to m th power k) rised to m th power E. Give our nswer in the form of n lgeric epression. ) Crlos is ers old t this moment. How old will Crlos e in 0 ers? ) An item in store costs dollrs. Wht is the price of the item, if fter discount, its price ws reduced to two thirds of the originl one? c) You hve dollrs to divide equll mong kids. How much will ech child get? d) You hve $00 to divide equll mong kids. How much will ech child get? e) Chrles ought more lmps for his prtment tod. If there re lmps in Chrles prtment now, how mn lmps were in his prtment efore the purchse? f) There re 0 ooks on ech shelf. How mn ooks re on shelves? g) There re students in clssroom. How mn students re still in the clssroom, if students leve? h) John is ers older thn Tom. If John is ers old, how old is Tom? E. Let d e vrile representing the distnce driven cr, nd let t represent the time it took to drive tht distnce. Write the following phrse s n lgeric epression: The distnce divided time. E. Let m e vrile representing the mss of given od, nd let represent its ccelertion. Use m nd to write the following phrse s n lgeric epression: The product of the mss of od nd its ccelertion. E. Let h e vrile representing the height of tringle, nd represent the se of the tringle. Use h nd to write the following sttement s n lgeric epression: One hlf of the product of the se of tringle nd its height. 0
13 E.6 Let. Rewrite the epression replcing the vrile with its vlue nd evlute, if possile. If evlution is not possile, eplin wh it is not possile. ) ) c) d) e) 6 f) E.7 If 0 the epression cnnot e evluted. Wh not? Cn e evluted when 0? Wht if? Find nother emple of n lgeric epression nd vlue of vrile(s) for which evlution is not possile. E.8 Let 0. Rewrite the epression replcing the vrile with its vlue nd evlute, if possile. If evlution is not possile, eplin wh it is not possile. ) ) c) d) 7 e) f) 0 E.9 Let. Rewrite the epression replcing the vrile with its vlue nd evlute, if possile. Otherwise, write undefined. ) ) c) E.0 Evlute A, if ) A ) A E. Sustitute 6 in the following epressions nd then evlute, if possile. Otherwise, write undefined. ) 8 ) 0 c) d) 6 e) 6 E. Sustitute in the following epressions nd then evlute, if possile. Otherwise, write undefined. ) ) c) d) e) 6 0
14 E. Sustitute 0 in the following epressions nd then evlute, if possile. Otherwise, write undefined. ) ) 00 c) d) e) f) E. Sustitute in the following epressions nd then evlute, if possile. Otherwise, write undefined. ) 000 ) 6 c) d) e) f) 6 E. Sustitute in the following epressions nd then evlute, if possile. Otherwise, write undefined. ) ) c) d) 7 e) f) E.6 Sustitute undefined. ) 0 ) 7 in the following epressions nd then evlute, if possile. Otherwise, write c) d) e) E.7 Sustitute in the following epressions nd then evlute, if possile. Otherwise, write 7 undefined. ) ) 7 c) d) 8
15 e) f) E.8 Sustitute undefined. ) in the following epressions nd then evlute, if possile. Otherwise, write ) c) e) d) 8 E.9 Sustitute 0. in the following epressions nd then evlute, if possile. Otherwise, write undefined. ). ).0 c) d) 0 e) 0. f) E.0 Sustitute 0. 6 in the following epressions nd then evlute, if possile. Otherwise, write undefined. ). ). 7. c) d) 600 e) 0.00 f) E Sustitute. in the following epressions nd then evlute, if possile. Otherwise, write undefined. ) ) 0. c) 0. d) 0. 0 e) E. If possile, evlute when ), 9 ), 7 c) 0.,. 08 f) 0 0.0
16 E. If possile, evlute when ), 9 ), 7 c) 0.,. 08 E. If possile, evlute when ), 9 9 ), 0 c) 0., 0. 0 E. If possile, evlute when ), ) , c) 0., 0 m E.6 If possile, evlute ( ) when ) 0, m 7 ), m c), m d) 0., m E.7 Use the letter to represent numer nd write the following sttements s lgeric epressions. Then evlute ech epression when. ) A numer douled ) Three fourths of numer c) A numer rised to the second power E.8 Evlute t, when t, t Bsed on our results, which of the following re true? ) t is lws positive ) t is lws negtive c) t m e positive or negtive depending on the vlue of t
17 Lesson Topics: Algeric epressions nd their evlutions; Order of opertions. Recll the order of opertions. The order of opertions When evluting rithmetic epressions, the order of opertions is ) Perform ll opertions in prentheses first. ) Do n work with eponents. ) Perform ll multiplictions nd divisions, working from left to right. ) Perform ll dditions nd sutrctions, working from left to right. If numericl epression includes frction r, perform ll clcultions ove nd elow the frction r efore dividing the top the ottom numer. Let e n unknown numer. If we increse the numer, the resulting numer cn e represented. Now, suppose tht fter dding, we multipl some other numer, clled, the result of the ddition. We m now write the epression ( ). Let us nlze wh we must plce prentheses round. If n lgeric epression involves more thn one mthemticl opertion, then the order of opertions is followed. Thus, if we simpl write (without using prentheses), we would onl e multipling nd, rther thn times the entire quntit. According to the order of opertions, using prentheses in ( ) indictes tht we dd to first, nd then multipl the result. Epressions nd ( ) hve entirel different menings, nd onl ( ) correctl represents the result of opertions performed in this emple. The order of opertions is used when lgeric epressions re evluted. For emple, let us evlute when. replce ech with rise to the second power 9 multipl it 8 sutrct from the result Emple. Rewrite the following epressions nd circle the rithmetic opertion together with its opernds tht hs to e performed first. Write the nme of the opertion net to our epression. ) )
18 ) We perform multipliction efore sutrction multipliction ) We eponentite efore multipliction eponentition Emple. List, ccording to the order of opertions, ll the opertions together with opernds tht re to e performed in the following epressions. ) ) 8 6 ) There re two opertions in, multipliction nd ddition. According to the order of opertions, multipliction should e performed first. Thus, the nswer is: Multipl, then dd. ) There re two opertions in 8, division nd multipliction. According to the order of opertions, the should e performed s the pper reding from the left to the right. Thus, the nswer is: Divide 8 first, nd then multipl. Emple. Write the lgeric epressions representing the following ) rised to the seventh power ) sutrcted from c) 8 times the quntit d) the opposite of Prentheses must e used in ech of these emples. 7 ) ( ) We lerned tht the eponent pertins onl to the closest 7 7 epression, so if we write insted of ( ), onl would e rised to the power 7. This rule cn e viewed s consequence of the order of opertions. 7 Eponentition should e performed efore ddition, hence in, is rised to the seventh power, nd then the result is dded to. To ensure tht is first dded to, we need to use prentheses, nd onl then the sum is rised to the seventh power. ) ( ) Notice the order of epressions: sutrct from indictes tht is written first nd we sutrct from it. c) 8( ) Thnks to prentheses, we dd first, nd then multipl. d) ( ) Tking the opposite mens multipliction. To ensure ddition first, we need prentheses. Emple. Use the letter to represent numer nd write the following s lgeric epressions. ) Add three to numer, nd then divide it z ) Seven more thn one third of numer c) A quntit decresed 9, nd then multiplied A d) A numer cued, nd then decresed
19 Plese, notice the use of prentheses in the emples elow. ) ( ) z or z ) 7 c) ( 9) A d) Emple. Determine if, in the following lgeric epressions, prentheses re necessr, i.e. the chnge or do not chnge the order of opertions. To this end determine if the first opertion tht should e performed is the sme s if the epression were written without n prentheses. ) ( ) c ) ( ) c ) In the epression ( ) c the first opertion tht should e performed is the ddition of nd. If we rewrite the sme epression ut without prentheses, we get c nd the opertion of ddition of nd is the first opertion s well (since ddition nd sutrction re lws performed in the order s the pper from the left to the right). Thus, we conclude, tht prentheses do not chnge the order of opertions in this cse. ) In the epression ( ) c the ddition of nd should e performed first, ut if we drop the prentheses the resulting epression is c nd thus we would first multipl nd c, nd then dd. Prentheses re needed since the do chnge the order of opertions. Emple.6 Rewrite the epression replcing vriles with their vlues. Then, evlute, if possile. If evlution is not possile, eplin wh it is not possile. ) if ) if nd m c) ( n ) if m nd n Ech vrile hs to e replced its given vlue nd the resulting numericl epression hs to e evluted. Plese, p ttention to the w prentheses re used. ), ut division zero is not defined. The epression 0 cnnot e evluted. The opertion is undefined. ) ( ) c) ( n ) m ( ) ( ) 7
20 Common mistkes nd misconceptions Mistke. When evluting 0, it is INCORRECT to write 0 0 ( 0 ) Insted, one should write 0 0 Numers or epressions not involved in the opertion tht is eing crried out must lws e rewritten. An equl sign mens tht the quntities on either side re equl. Eercises with Answers (For nswers see Appendi A) E. Write the lgeric epression representing the following. ) m n sutrcted from ) the opposite of m n c) m n multiplied 7 d) sutrcted from m n e) the opposite of k k f) divided (use the smol in our nswer) E. Write the following phrses s lgeric epressions. Rememer to plce prentheses where needed (plese, plce them onl when needed). ) Multipl, nd then dd ) Multipl the sum of nd c) The opposite of, then rise it to the sith power d) Sutrct from, nd then multipl the result z e) Rise to the third power, nd then multipl the result 9 f) Multipl 9, nd then rise the result to the third power g) The difference of nd, then divided c h) Divide, nd then dd i) The opposite of the sum of M nd j) Rise to the third power, rise to the seventh power, nd then dd them together E. Use the letter to represent numer nd write the following s n lgeric epression. ) A numer decresed 7, nd then douled ) Add c to numer, nd then tke two thirds of the sum c) Tke one fourth of numer, nd then sutrct from it d) Multipl numer 9, nd then sutrct it from c e) A numer, first divided, nd then rised to the third power. f) The opposite of numer, then multiplied g) A quntit rised to the third power, nd then incresed 6 h) A numer decresed, nd then the result multiplied i) Sutrct numer from nd then tke the opposite of the result 8
21 j) A numer multiplied the sum of the sme numer nd k) The opposite of numer, then rised to one hundred nd twent first power l) Squre numer, nd then tke the opposite of it E. Let C e vrile representing the temperture in Celsius. Write the following phrse s n lgeric epression: Nine fifths of the Celsius temperture plus. E. Let L e vrile representing the length of rectngle, nd W its width. Use L nd W to write the following phrse s n lgeric epression: The sum of the length of rectngle nd its width, then multiplied. E.6 Let m represent mss nd c the speed of light. Use m nd c to write the following phrse s n lgeric epression: The product of mss nd the squre of the speed of light. E.7 In the following epressions circle the rithmetic opertion, together with its opernds, tht hs to e performed first. Write the nme of the opertion net to our epression. For emple, in, multipliction of nd hs to e performed first, thus the nswer is + multipliction ) ) ( ) 8 c) 8 d) ( ) e) c f) c g) 7 h) E.8 There re two opertions in the lgeric epression, ddition nd multipliction. In order to evlute, we would hve to perform them ccording to the order of opertions. First multipl nd, nd then dd. List, ccording to the order of opertions, the opertions tht re in the following lgeric epressions. ) ) s c) ( ) d) t e) f) ( ) g) ( c) h) c E.9 Determine if, in the following lgeric epressions, prentheses re necessr, i.e. the chnge or do not chnge the order of opertions. To this end, determine if the first opertion tht should e performed is the sme s if the epression were written without n prentheses. If the opertion is different, write prentheses re needed, otherwise rewrite the epression without n chnges. ) ( ) ) ( c ) c) ( ) d) ( c ) ( c d) e) ( ) f) 9
22 g) i) ( ) h) 8 () () j) ( d) c E.0 Evlute, if possile. ), if ), if c), if d) Did ou get the sme nswer for,, nd c? Cn ou eplin wh it is so? e) If when, evlute when. You should e le to rrive t our nswer without performing n evlution. E. Let. Rewrite the epression replcing the vrile with its vlue nd evlute, if possile. Otherwise, write undefined. ) ) c) d) ( ) e) f) g) E. Let. Rewrite the epression replcing the vrile with its vlue nd evlute, if possile. Otherwise, write undefined. ) ) ( ) c) ( ) d) E. Let. Rewrite the epression replcing the vrile with its vlue nd evlute, if possile. Otherwise, write undefined. ) ) c) ( )( ) d) e) 0
23 E. Sustitute undefined. ) A c) A nd then evlute the following epressions, if possile. Otherwise, write ( A) d) ) A A E. Let 0.. Rewrite ech epression replcing the vrile with its vlue nd evlute, if possile. Otherwise, write undefined. ) ) c) 0. d) E.6 The epression cnnot e evluted for which of the following vlues of nd? Eplin wh. ), ), c), d), e), 0 f) 0, E.7 If possile, evlute when m, n. Otherwise, write undefined. Before evluting, rewrite the epressions sustituting the numericl vlues of m nd n. ) m n ) m( n) c) ( m n) d) ( m ) n e) ( m ) n E.8 If possile, evlute when m, n. Otherwise, write undefined. Before evluting, 8 rewrite the epressions sustituting the numericl vlues of m nd n. ) 8m 0n ) 0 mn c) ( n m) d) 8 m n e) n ( m) 8 f) n m 0 A
24 E.9 If possile, evlute when A, B. Otherwise, write undefined. Before evluting, rewrite the epressions sustituting the numericl vlues of A nd B. ) A ) B c) B A B d) A B A( B) e) A B E.0 Evlute the following epressions: n, m. If evlution is not defined, write undefined., n,, ( ), n, nm if,, E. Let, 0., nd z. If possile, evlute the following epressions. Otherwise, write undefined. ) ( z ) ) z E. Let 0., 0., c. If possile (otherwise, write undefined ), evlute the following epressions. Before evluting, rewrite the epressions sustituting the numericl vlues of vriles. ) c ) c c) 0 E. Find the vlue of ) ) ( ) if E. Find the vlue of A B, if ) A, B ) A, B c) A 0., B 0. 7 d) A, B 8 e) A, B f) A, B E. Find the vlue of ( A B), if ) A, B ) A, B c) A 0., B 0.
25 d) A, B e) A, B 7 6 f) A, B 9 0. E.6 Find the vlue of, if ), 0. 0 ) 00, 0. c) 0., 0. E.7 Evlute the following epressions, if m, n, nd p. Before evluting, rewrite the epressions sustituting the numericl vlues of vriles. ) m ( n p) ) m n p E.8 In the following epressions, identif the first opertion tht should e performed ccording to the order of opertions nd ntime it is numericl opertion, perform it. ) ) c) () d) ( ) e) f) E.9 Write the following phrses s lgeric epressions nd then evlute them when. ) multiplied, nd then squred ) sutrcted from, nd then divided 0. c) 9 divided, nd then cued
26 Lesson Topics: Equivlent lgeric epressions. Definition of equivlent lgeric epressions Suppose we wish to write s n lgeric epression numer douled. Should we write or? Becuse of the commuttive propert of multipliction, oth nswers re right. Both hve the sme mening, lthough the pper to e different. We encounter similr ide in rithmetic. The frctions nd re equivlent, which mens tht the represent the sme numer lthough the do not look the sme. Similrl, we would s tht nd re equivlent (we often s equl) nd write. Equivlent Algeric Epressions Two lgeric epressions re equivlent if, when evluted, the hve the sme vlue for ll replcements of the vriles. Suppose tht two lgeric epressions re equivlent, like the two mentioned ove, nd. Wht it mens, ccording to the definition, is tht if we choose n vlue of, let s s, nd evlute, nd then evlute, the results must e the sme. If we chnge the vlue of, for emple to, gin two results re equl ( 8, nd 8 ). No mtter wht the vlue of, the two results re lws going to e equl. Thus, to determine tht two epressions re equivlent one would hve to evlute them for ll possile sets of vlues of vriles. Since we cnnot check ll, we cnnot prove equivlence performing evlution (mke sure tht ou understnd tht even if we determine tht two epressions ssume the sme vlue for mn sets of vlues of vriles, we still cnnot clim tht the two epressions re equivlent). To prove the equivlence of lgeric epressions, some generl rules must e emploed. Terms nd fctors In rithmetic, we often refer to numers tht re eing dded s terms, nd to numers multiplied s fctors. For emple, nd re terms of ddition 7, while nd re clled fctors of, since. We will now generlize the notions of terms nd fctors. Terms Algeric epressions tht re dded (or sutrcted) re clled terms. Ech sign, + or, is prt of the term tht follows the sign. In other words, the ddition nd sutrction signs rek the epression into smller prts, clled terms, nd so, in there re three terms:,,. Notice tht ecuse is preceded minus sign, the minus sign is prt of the term:. The epressions nd re terms in. d d Some epressions hve just one term. For emple, oth nd hve onl one term.
27 Fctors Algeric epressions tht re multiplied re clled fctors. The epression mn hs fctors,, m, n nd n comintion of those, like m, n, nd of course, mn. In, the epressions nd re clled fctors ut,,,, re lso fctors of this epression. During our stud we will e tlking out eplicit fctors. Eplicit fctors of mn re, m nd n, i.e. the fctors tht re seprted the multipliction sign (displed or not displed) of n epression. Eplicit fctors of re nd. The epressions nd re eplicit fctors in ( )( ). Emple. List ll terms of the following epressions. ) z ) ( z ) ) The terms re,, z. Rememer tht signs re lws prt of terms. We list s term ( is preceded the minus sign). ) The terms re,( z ),. Notice the minus sign in nd, nd tht the epressions ( ) should e viewed s one term. Emple. List ll eplicit fctors of the following multipliction. ) ) ( ) ) Since, the eplicit fctors re,,. ) Since ( ) ( ), the fctors re;, ( ) nd. Notice tht when listing fctors, the epressions in prentheses re treted s one unsplitle epression. Alger is n strct generliztion of rithmetic, where numers re replced with vriles. The lws tht re true for numers lso hold for lgeric epressions (recll, lgeric epressions re merel smolic representtions of numers). We will discuss some of the lws in the contet of equivlent epressions. Commuttive propert of ddition: rerrnging terms results in equivlent epressions We know tht nd re oth equl to the sme numer, 8. It is ecuse the result of ddition does not depend on the order of numers tht re eing dded. This propert is clled the commuttive propert of ddition. Rememer, sutrction does not hve this propert:. But, if we view sutrction s the ddition of the opposite numer, we get ( ). With the use of vriles, we cn epress the ove ides in generl form (without the use of specific numers). For n vlue of nd,
28 Commuttive Propert of Addition Also, since ( ), we hve Consequence of Commuttive Propert of Addition Equivlentl, we cn s tht, chnging the order of terms results in n equivlent epression (n epression tht looks different, ut mens the sme). For emple, The terms of re nd. If we reverse the order of terms, we otin. The terms of re nd. Reversing their order gives us. The terms of c d re c nd d. Reversing their order gives us c d d c. Similrl, if n lgeric epression consists of more thn two terms, we cn rerrnge them in n order. For instnce, the terms of B C re, B, nd C. Thus, we cn rewrite the epression B C s B C or B C : B C B C B C (there re more possile rerrngements). Emple. Using the fct tht chnging the order of terms results in n equivlent epression, rewrite the following epressions in their equivlent form rerrnging the terms. Use the equl sign to indicte tht the resulting epressions re equivlent. ) 7 ) ( c) ) The terms re 7 nd. We get 7 7 ) The terms of ( c) re ( c) nd. If we reverse the order, we get ( c) ( c ) Emple. Determine which of the following epressions re equl to z. ) z ) z c) z All of them re. The terms of z re, nd z. As long s the sign tht is in front of n epression is not ltered, we cn rerrnge terms. The sign tht goes efore is plus, nd in ll epressions ()(c) is lso preceded plus sign. follows the minus sign nd the sme is true for ll epressions ()(c). Finll, z is preceded plus sign in ll these epressions. 6
29 Emple. Replce with epressions such tht the resulting sttements re true. Use prentheses when needed. ) z ) z z ) z z ; z ) z z() ; () Commuttive propert of multipliction. Rerrnging fctors results in equivlent epressions Multipliction, like ddition, is commuttive. the result of multipliction does not depend on the order,. In generl, we hve Commuttive Propert of Multipliction or This mens tht, the rerrngement of the order of fctors results in n equivlent epression. For emple,, or ( ) ( ). Fctors cn lso e rerrnged if we hve more thn two fctors. For emple,, cdef dcef efdc fedc (or n other order of fctors of c, d, e, nd f) Emple.6 Rewrite the epression different order. in two equivlent forms multipling its fctors in We cn rewrite the ove epression in more thn two equivlent forms. For emple,,,,, or. An two would e the correct nswer. Appling rules of opertions on frctions results in n equivlent epression The rules for ddition nd sutrction of frctions with common denomintors re Rule for Addition nd Sutrction of Frctions with Common Denomintors c c c, c 0 c c c 7
30 For emple, nd thus lso (notice, tht we switched the left side with the right one of the eqution, so oth equtions re true, just like if then ). It is lso true tht nd. The rules for multipliction nd division of frctions: Rule for Multipliction nd Division of Frctions, 0 Thus, we should recognize tht the following re true:,, nd ( ) Finll, ou might recll tht would like to see how to do it) tht. Similrl, one m show (sk our instructor, if ou Rule for Negtive Signs in Frctions, 0 Emple.7 Rewrite ech of the following epressions s sum or difference of two epressions. Use or (we ssume tht c 0 ). Indicte with the c c c c c c equl sign tht the resulting epressions re equivlent. s ) ) ) ) s s 8
31 Emple.8 Use the fct tht to rewrite the following epressions s product of numericl fctor nd n lgeric epression. ) ) ) ; is numericl fctor. ) ( ) Notice the use of prentheses. The entire epression in the numertor should e multiplied. Hd we omitted the prentheses, onl would hve een multiplied. Emple.9 Rewrite ech of the following epressions in their equivlent form s single frction. 6 ) ) c) ( ) d) c e) f) s s 6 6 ) ) ( ) c) ( ) Write insted of ( )! d) e) First replce, nd s s s s s s s (equls cn e sustituted equls). Then sutrct, just s ou sutrct s s frctions. c ( ) c ( ) ( c) f) Notice the use of prentheses. 9
32 Emple.0 Determine which of the following epressions re equl to, c c c c, ( ), c. c The onl epression tht is not equivlent is c (since ). To demonstrte tht, let us set 0, c nd evlute 0, c 0. Since, we conclude tht is not equivlent to. c c c The epression c is equivlent ecuse of the rules for ddition of frctions. c ( ) is equivlent ecuse of the rules for multipliction of frctions. is c c equivlent ecuse we cn replce its equivlent epression. Performing numericl opertions results in equivlent epressions There re mn opertions one cn perform on n lgeric epression to otin n equivlent one. One of them is performing numericl opertions ccording to the order of opertions. For emple, 6 Emple. When possile, perform numericl opertion to crete n equivlent epression. If no numericl opertion cn e performed, clerl indicte so. ) ) c) 8 d) m ( ) e) g) ( ) h) m f) ( ) i) ( m)( n) 6 ) ) Divide the numertor nd denomintor to reduce the frction. 8 c) Since multipliction of nd hs to e performed efore ddition, no numericl opertion cn e performed. m m d) ( ) 6 m e) Since must e first rised to m th power, no numericl opertion cn e performed. 0
33 f) ( ) ( ) 6 g) ( ) ( )( ) h) 6 We cn cncel in the denomintor with the fctor of in 6. i) ( m)( n) m( ) n ( ) mn 6mn How to show tht two epressions re not equivlent Two lgeric epressions re equivlent, if for ll vlues of vriles the ssume the sme vlue. Thus, if we cn find just one set of vlues of vriles for which the epressions do not ssume the sme vlue, it is enough to conclude tht the re not equivlent. As n illustrtion, we will demonstrte tht is not equivlent to. To this end, we must find some vlue of tht when evluted, the two epressions ssume different vlues. We will use (the choice of the vlue of is ritrr). We evlute oth lgeric epressions. nd 0. Since 0, we conclude tht is not equivlent to. Notice tht there re other vlues of for which is not equl to, ut since we onl need one such vlue, we lred proved tht in not equivlent to. Emple. Show tht the following two epressions ( ) nd re not equivlent evluting them when nd demonstrting tht the vlues of the two epressions re not equl. () () 8 ut. Since 8, not equivlent. ( ) nd re Common mistkes nd misconceptions Mistke. s When writing n epression like, for emple,, s single frction, mke sure tht ll the m m smols re clerl ove the frction r. You should not write The nswer should e written s s, so the minus sign is included in the numertor. m
34 Eercises with Answers (For nswers see Appendi A) E. Write word to complete ech sentence. In the epression, nd re clled. In the epression, nd re clled. E. List ll terms of the following epressions. ) ) cd c) d) ( ) z E. Is +8 equl to 8+? Is 8 equl to 8? How out cd cd nd? Wh? E. ) Evlute m n nd n m when m nd n. Bsed on this evlution, cn ou determine if the two epressions re equivlent? ) Is it true tht m n n m? E. For ech of the following epressions  List ll its terms  Using the fct tht chnging the order of terms results in n equivlent epression, rewrite the following epressions in their equivlent form rerrnging the terms. Use the equl sign to indicte tht the resulting epressions re equivlent (for emple, the epression A 9 should e rewritten s A 9 9 A). ) m z ) c) c d) e) c( d f ) f) s ( ) E.6 Fill in the lnks to mke true sttement. ) mn mn ) ( ) E.7 List ll terms, nd then, chnging the order of these terms, crete two new equivlent epressions for ech of the following. ) ) c E.8 For ech of the following epressions ()() find n epression equivlent to it mong epression (A)(E). Rewrite ech mtched pir with the equl sign etween them to indicte their equivlence. () s t u (A) t u s () t s u (B) s t u () u s t (C) t s u () u t s (D) s u t () s t u (E) s u t
35 E. 9 Rewrite the following epressions plcing the multipliction sign ' whenever (ccording to the convention) it ws omitted. Then, identif ll eplicit fctors. ) ) ( ) c) d) ( )( c) E.0 Rewrite ech of the following epressions in its equivlent form using. Use the equl sign to indicte tht the resulting epression is equivlent to the originl one (for emple, the epression 9A should e rewritten s 9A A9 ). Rememer out prentheses. ) mn ) 7 c) cd d) c( d) E. ) Rewrite the epression vst in its equivlent form chnging the order of its fctors to crete three new equivlent epressions. Indicte their equivlence using the equl sign (for emple, one of the nswers might e vst tsv ). ) Repet the ove eercise for v ( ) t. E. ) Is AB equivlent to BA? How out AB nd BA, AB nd BA ()? Wh? ) Is AB equivlent to BA? Wh? Support our nswer evluting oth epressions when A nd B. E. Is equivlent to? How out nd? Wh? How out ( mn )( ) nd ( )( nm)? Wh? E. According to the rules for dding nd sutrcting frctions, we hve c (ssume c 0 ) Rewrite ech of the epressions elow s sum or difference of two c c c epressions. Use equl signs to indicte tht the resulting epressions re equivlent to the originl ones (for emple, the epression ) c) 7 E. Using the fct tht nd c c c epressions s single frction. Do not simplif. n ) m n c) c c t t should e rewritten s t ). 6 ) c d) c c m 7m n ) d) A B C c c c c nd (ssume c 0 ), rewrite the following
36 E.6 Write the following epressions s single frction using the fct tht 7 ) 7m n ) m t c) s s s d d c d c d E.7 According to the rule for multipliction of frctions the following is true:. We cn 7 7 s tht the quotient ws written s product of numericl fctor nd n lgeric epression 7 7. Write the following epressions s product of numericl fctor nd n lgeric epression. ) ) c) e) ( ) d) f) E.8 Using the rule for multipliction of frctions, rewrite ech of the following epressions in their equivlent form s single frction (for emple, ). Rememer to use prentheses when needed. m m ) ) n n c) d) t e) ( s ) f) n n m n g) h) t E.9 Students were to write n nswer to the following prolem: Using lgeric smols, write n opposite numer to t s. Student A gve the nswer: Student B gve the nswer: s t s t
37 Student C gve the nswer: Who ws right? Wh? s t E.0 You know tht. B plcing the minus sign differentl, write ech epression in two dditionl equivlent ws. Use prentheses when needed. c ) ) d E. Write ech of the following epressions in their equivlent form s single frction. Do not simplif. ) ) c) 7 d) t t n e) f) t t k t k t cd m n g) h) t t E. Fill in the lnks to mke true sttement. ) c) ) d) E. Perform ll numericl opertions tht re possile. If none re possile, write not possile. ) 8 ) ( 8) 8 c) d) 8 e) 8 f) g) ( ) h) ( ) m i) ( ) j) m k) 0.(0 ) l) m) 0 n) (6 ) o) p) z d q) r) 9c
38 s) ( 0.)( 0.z) t) E. Is equivlent to? Is equivlent to? E. Is ( )( ) equivlent to ( )? Eplin our nswer. E.6 Replce with epressions such tht the resulting sttement is true. Use prentheses when needed. ) c c ) c) d) e) f) z c c g) z z h) E.7 Determine which of the following epressions re equivlent to m m m,, m (), m, m E.8 Determine which of the following epressions re equivlent to m n : n m, m( n), m (n), nm, n m, ( )n m E.9 Determine which of the following re equivlent to c d. c d, c d, d c, c d E.0 Determine which of the following re equivlent to,,,,., E. Determine which of the following epressions re equivlent to, 0,,,, E. Determine which of the following epressions re equivlent to.,,,,, 8 6
39 E. Determine which of the following epressions re equivlent to, 8, 8,, 8. 8 E. Determine which of the following epressions re equivlent to 6, ( ), 6, 6, 6 6, 6 6 ( ). 6 m n E. Determine which of the following re equivlent to. n m m n,, m n, ( m n), n m E.6. The correct nswer to prolem is Mr whose nswer is t v? vt. John s nswer is vt. Is John right? How out E.7 Show tht ( ) is not equivlent to evluting oth epressions when nd demonstrting tht the vlues re not the sme. E.8 Show tht is not equivlent to ( ) evluting oth epressions when, nd demonstrting tht the vlues re not the sme. E.9 Show tht m n p is not equivlent to m ( n p) evluting oth epressions when m, n, nd p nd demonstrting tht the vlues re not the sme. m m E.0 Evlute ( ) nd when ) m ) m c) m d) m 7 e) Bsed on the ove, cn ou determine if f) Evlute equivlent? m ( ) nd m when m m ( ) nd m re equivlent?. Cn ou now determine if m ( ) nd m re 7
40 Lesson Topics: Opertions on power epressions with nonnegtive integer eponents. Eponentil nottion Eponentil nottion is used to write repeted fctors in compct w. Eponents re nother w of writing multipliction. For emple,,. times times We will etend this ide to lgeric epressions. Eponentil Epression The epression of the form n is clled n eponentil epression. It is defined s follows 0,,, n..., for n =,,,. n times ( is repeted s fctor n times). is clled the se, n is clled n eponent or power. Notice tht, ccording to the definition n epression rised to the zeroth power is equl to one. Also, vrile tht ppers to hve no eponent is rised to the first power. = n Recll tht is red s to the nth power. In the cse of, often, insted of to the second power we red it s squred ; is often red s cued. You should lso rememer tht: The eponent pertins onl to the closest numer or vrile. To ppl the eponent to the entire epression we must plce prentheses round the epression. The eponent is then plced outside the prentheses. For emple, in, onl is rised to the third power. in ( ), is rised to the third power. or in in n, onl is rised to the nth power (n pertins onl to not to ) n ( ), is rised to the nth power 8
41 The ove convention cn e interpreted s the consequence of the order of opertions. For emple, in ( ) prentheses indicte tht multipliction should e performed first, nd onl then the result is rised to the second power. This mens tht the entire is rised to the second power. Without prentheses, we first eponentite, nd then multipl, so onl is squred. Notice lso, tht without this convention, we would hve no mens to distinguish etween, let s s, nd ( ), or ( ) nd. Emple. Epnd, tht is write without eponentil nottion. ) ( A ) ) A ) The eponent pertins to the entire epression A: (A) AAA ) The eponent pertins onl to A : A A A A. Emple. Rewrite using eponentil nottion whenever it is possile. ) 7 mmm ) c) ( c d) (d c) d) ) 7mmm 7m tomes ) times times c) Notice tht c d d c, thus ( c d) (d c) ( c d)( c d) ( c d times times d) Notice tht is equivlent to, thus ) Emple. Evlute ) 0 ( ) ) 0 c) ( mn 7 ) 0 ) Rememer tht n epression rised to the zeroth power is equl to, nd tht ecuse of prentheses the zero power pertins to the entire epression, thus () 0. ) This time onl is rised to the zeroth power, hence c) The epression m n is rised to the zeroth power, so ( mn ) 9
42 Numericl coefficients Numericl Coefficient In product of numer, vriles or lgeric epressions, the numericl fctor is clled numericl coefficient. For emple, in the epression, the numericl coefficient (often clled the coefficient) is equl to. m The coefficient of ( ) is nd the coefficient of is. If there is no numer in front of product, it is implied tht the coefficient is. If there is negtive sign, it is implied tht the coefficient is. For instnce, the coefficient of is equl to (ecuse ). The coefficient of is (ecuse ). Emple. In the following epressions, identif ses, eponents nd numericl coefficients. ) 7 ) ) se: ; eponent: 7; numericl coefficient: z k 9 ) se: k z ; eponent: 9; numericl coefficient: c) se: ; eponent: ; numericl coefficient: c) Lws of eponents Consider ( ) ( ) ( ) ( ) We hve s nd nother s for totl of s (to multipl eponents). The ide cn e generlized to otin nd we dded Product Rule for Eponents m n mn When multipling eponentil epressions with like ses, we dd the eponents nd keep the 0 0 common se. For emple,. If we hve Out of repeted s in the numertor, we cnceled s for totl s (to divide we sutrcted eponents). In generl one cn prove tht 0
43 Quotient Rule for Eponents m n mn, 0 When dividing eponentil epressions with like ses, sutrct the eponents nd keep the 0 common se. For emple, 6 If we hve ( ) ( ) ( ) We cn think of this s groups of for totl of 6 s. In generl, Power Rule for Eponents ( m n mn ) When rising n eponentil epression to nother power, keep the sme se, nd multipl the eponents. For emple,. Now, consider This cn e etended to ( ) ( ) Product to Powers Rule for Eponents ( n n ) n An eponent outside the prentheses pplies to ll prts of product inside the prentheses nd thus to rise product to power, one cn equivlentl rise ech fctor to tht power. For emple,. Finll,, 0 In generl, Quotient to Powers Rule for Eponents n n n, 0 An eponent outside the prentheses pplies to ll prts of quotient inside the prentheses. Thus, to rise quotient to power, one cn equivlentl rise oth numertor nd denomintor to tht power. For emple,.
44 For our convenience, we will displ ll the rules together. Lws of Eponents. 0.. m n mn ( ) 6.. m n mn. n n n n n ( ) 7. n m n mn, 0, 0 Emple. Perform the indicted opertions nd simplif. 8 ) ( )( ) ) c) ( 6 ) 6 7 d) 8 ( ) e) ( )( ) f) ( ) ) ) c) ( )( ) ( )( ) ( ) ( ) d) e) f) ( )( ) ( ) 7 ( ) ( ) ( ) 7 ( ) Emple.6 Rewrite in its equivlent form s single eponentil epression without prentheses. Identif the numericl coefficient of the finl epression. ) ( ) ) ( ) ) ( ) ( ) ( ) ; the coefficient: ) ( ) ( ) ; the coefficient: 6. Emple.7 Write s n lgeric epression using prentheses where pproprite, then remove the prentheses nd simplif. ) The product of nd, then rised to the sith power ) The quotient of nd, then rised to the second power
45 ) ) ( )( ) ( ) ( ) ( ) ( ) ( ) () 6 ( Emple.8 Simplif the epression ) nd then evlute, when. 0 ( ) ( ) To evlute, we sustitute ( ) Emple.9 Which of the following epressions re equivlent to 7, ( 7 ), ( ), 7, 7? ( ) 0, nd hence ( 7 equivlent. is equivlent ecuse onl the order of fctors chnged ( ), 7 7 7, 7. ) is not equivlent. All others re 7 nd Emple.0 Find the numericl vlue of such tht the following sttements re true. ) 6 7 ) hs een 6 ) We must epress s n eponentil epression with the se (to mtch it 6 6 to ). Since, we get ( ). As result, hence. ) We must epress Notice tht 6 6, nd thus s n eponentil epression with the se (6 ) ;. 8 9 Emple. Evlute. ) )
46 ) ) 6. Common mistkes nd misconceptions Mistke. There is difference etween squred, Mistke. In the epression nd ( ). In ) ( )( ). Just like 9 ( Mistke. Although it is true tht We recognize tht onl is squred, in, while ( ) 9. 7, since the ses re not the sme, DO NOT dd eponents. ( ), ( 7 ) 9, however, ( ). ( 7 ) 9. ( ), is Mistke. Plese, rememer 8 (not 7 ). In other words, if one of the fctors does not hve n eplicit eponent, it mens it is rised to the first power, nd thus one hs to e dded. Eercises with Answers (For nswers see Appendi A) m E. In the epression, is clled the, m is clled the or nd is clled the. E. E. Fill in the lnks. ) An epression rised to the power is equl to itself. ) An epression rised to the power is equl to. m ) In the epression the eponent pertins to. n ) In the epression ( ) the eponent pertins to. c) In the epression d) In the epression e) In the epression c ) n ( de the eponent pertins to. n () the eponent pertins to. n f) In the epression the eponent pertins to. m the eponent pertins to.
47 E. E. E.6 Write the following sttements s lgeric epressions using prentheses where pproprite. ) The quotient of nd, then rised to the fourth power ) Two thirds of, then rised to the third power c) c cued, nd then divided 7 d) The product of nd, then rised to the second power e) Rise to the second power, nd then multipl the result 8. f) The opposite of, then rised to the fifth power. g) Rise to the tenth power, nd then tke the opposite of the result. In ech of the following epressions, identif the se, eponent nd numericl coefficient. m ) ) c) d) ( c) e) g) m 7 z w f) h) ( ) 7 ( ) Write using eponentil nottion whenever it is possile. ) ) zzzz c) d) e) f) g) ( )( )( ) h) (t )(t )(t )(t ) kkk i) m nn m j) k n z z z ( z)( z)( z) k) l) zzzz z z z m) n) ( )( )( ) o) p) q) r) ( w v)(v w)(v w) m m m m s) ( m n)( m n) m n t) ( m p n)( p m n)( n m p) E.7 Write the following epressions without using eponentil nottion. ) ( ) ) c) e) g) ( m) d) ( ) f) ( ) h) m
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