Introduction to Algebra  Part 2


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1 Alger Module A Introduction to Alger  Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008
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3 Introduction to Alger  Prt Sttement of Prerequisite Skills Complete ll previous TLM modules efore eginning this module. Required Supporting Mterils Access to the World Wide We. Internet Eplorer. or greter. Mcromedi Flsh Pler. Rtionle Wh is it importnt for ou to lern this mteril? Alger is tht rnch of mthemtics tht mkes use of smols to represent unknown quntities. An understnding of how to mnipulte lgeric equtions llows the student to solve for mn unknown quntities. A solid grsp of lger is essentil if the student wishes to succeed in other res of mth such s trigonometr nd clculus. Lerning Outcome When ou complete this module ou will e le to Solve prolems using lger. Lerning Ojectives. Use the lws of eponents to simplif lgeric epressions.. Perform lgeric division.. Work with zero nd negtive eponents.. Perform the opertion of simplifiction on frctions hving more thn one term in the numertor.. Perform lgeric long division. Module A Introduction to Alger: Prt II
4 OBJECTIVE ONE When ou complete this ojective ou will e le to Use the lws of eponents to simplif lgeric epressions. Eplortion Activit EXAMPLE Simplif the following: ) ( )( ) ( )( ) 7 ) ( )( ) ( )( ) 7 This is tedious process nd shorter method cn e used which is shown in Emple. EXAMPLE Simplif the following: ( )( ) 7 however 7 This rings us to the rule for multipling which is: ( m )( n ) mn tht is, we keep the se, nd we dd the eponents m nd n. Another form of multipliction is to rise power of numer to power. Module A Introduction to Alger: Prt II
5 EXAMPLE Simplif the following: ) ( ) ( )( )( )( ) ) ( ) 6 However this cn e done the following rule for multipliction: ( m ) n mn tht is, we multipl the powers, i.e. ( ) 8 NOTE: It is ver importnt to rememer the difference etween the rules: RULE : ( m )( n ) mn multipling like ses RULE : ( m ) n mn rising power to power EXAMPLE Simplif: ) ( )( ) ) ( 7 ) c) ( ) Wh? (Smol mens not equl to) Module A Introduction to Alger: Prt II
6 EXAMPLE Simplif the following: ) ( )( ) regroup the terms to otin ppl rule for multipling eponents 6 ) ( )( c) regroup the terms to otin c ppl rule 6 c c) ( )( ) regroup terms to otin ppl rule 6 There is one more importnt rule we must consider, when rising power to n eponent: RULE sttes: ( p q ) r pq qr Module A Introduction to Alger: Prt II
7 EXAMPLE 6 Simplif the following: ) () epnd ()()()() regroup OR ppl Rule ove: () ) ( ) epnd ( )( )( ) regroup 6 6 OR ppl Rule ( ) 6 6 OR keep the rckets in the first step if ou prefer ( ) ( ) ( ) 6 6 Module A Introduction to Alger: Prt II
8 c) ( ) ppl Rule 8 d) (c d ) ppl Rule () c d note: the must lso e squred 6c d 6 e) ( ) ppl Rule Module A Introduction to Alger: Prt II
9 Eperientil Activit One Perform the indicted multiplictions.. (n ) 6. ( ). ()( ). (ct )(9t ). ( )(7 )( ) 6. (8dr s )(drs ) 7. (st ) 8. (t 7 ) Show Me. 9. ( ) 0. (7πR ) Eperientil Activit One Answers. n ct d r s 6 7. s t t 8 (TLM 8 t 8 ) π R 6 Module A Introduction to Alger: Prt II 7
10 OBJECTIVE TWO When ou complete this ojective ou will e le to Perform lgeric division. Eplortion Activit In lger, division of one quntit nother is most commonl written s frction: Onl fctors common to oth numertor nd denomintor cn e divided out. EXAMPLE Simplif the following: ) ) 6 c) 6 the is common to oth the numertor nd the denomintor so it is divided out the is common, so it divides out to leve the the common fctor is so it is divided out 6 d) the common fctor is, so when it is divided out we re left with As in multipliction, lgeric division uses rules of eponents to simplif epressions. 8 Module A Introduction to Alger: Prt II
11 EXAMPLE Simplif the following: ) Epnd: Simplif dividing the two s in the denomintor into two s in the numertor, ) 6 epnd: simplif dividing the three s in the denomintor into three s in the numertor, to e left with Agin, this is tedious process nd shorter method cn e used. This shorter method is shown in Emple. Module A Introduction to Alger: Prt II 9
12 EXAMPLE Simplif the following: ) However, this cn e done the following rule for division: RULE : m n mn ver importnt rule! Thus or just. ) c c 6 ppl rule. c 6 c c) z z epnd: ppl rule z  6 z z z 0 Module A Introduction to Alger: Prt II
13 Module A Introduction to Alger: Prt II d) epnd: ppl rule. 6
14 Eperientil Activit Two Perform the indicted divisions t u t 6. rt s 7ts 8. 8 n n 9 c ds c 9 z 6w z 6 w Show Me. Eperientil Activit Two Answers.. n.. c ds. t u z rt s 8. w Module A Introduction to Alger: Prt II
15 OBJECTIVE THREE When ou complete this ojective ou will e le to Work with zero nd negtive eponents. Eplortion Activit RULE : 0 here 0 i.e. n quntit rised to the zero power is equl to. REASON: ecuse we re dividing something into itself, nd division, therefore, 0 since oth 0 nd re equl to. 0 from the rule for EXAMPLE Oserve the one in ech of the following epressions: ) 0 ) 0 c) 0 ( 0 ) () d) (8) 0 e) 0 f) () 0 RULE 6: n n 0 Module A Introduction to Alger: Prt II
16 EXAMPLE ) ) c) d) since onl hs the eponent of e) f) HINT: A negtive eponent on quntit moves tht quntit from numertor into denomintor, or from denomintor into numertor. eg., here is moved from the numertor to the denomintor. eg., here is moved from the denomintor to the numertor. Of course the eponent goes long with the quntit, onl its sign chnges to positive. Module A Introduction to Alger: Prt II
17 EXAMPLE Simplif the following: ) 7 ppl the rule for dividing with eponents 7 eliminte the negtive eponent ppling rule 6: ) epnd: ppl the rule for dividing with eponents: eliminte negtive eponents ppling rule 6: c) eliminte negtive eponents ppling rule 6: NOTE: The nd the ech hve eponents of. The negtive eponents onl ppl to the vriles. Think of the question s the following: The nd hve positive eponents nd st where the re!!! Module A Introduction to Alger: Prt II
18 z d) z epnd: z z ppl the rule for dividing eponents z z eliminte negtive eponents rewriting individul fctors of the term nd get: nd z z nd then sustituting to get: z e) simplif ( ) 6 eliminte negtive eponents ppling rule 6: 6 6 Module A Introduction to Alger: Prt II
19 The Rules of Eponents... ( ( ( m )( n ) m n mn ) l m n ) ln m n mn. m n mn. 0 m 6. m Module A Introduction to Alger: Prt II 7
20 Eperientil Activit Three Epress the following with onl positive eponents. PART A c 6. ( ) ( ) PART B Simplif the following nd epress our nswers with onl positive eponents (. 6. ) 7. 6 c c 8 7 m n mn rst ( r s t) ( ) 8. ( ) 9. ( z ) 0. ( ) Show Me. 8 Module A Introduction to Alger: Prt II
21 Eperientil Activit Three Answers PART A c 8. PART B c z m n 7 s t r Module A Introduction to Alger: Prt II 9
22 OBJECTIVE FOUR When ou complete this ojective ou will e le to Perform the opertion of simplifiction on frctions hving more thn one term in the numertor. Eplortion Activit EXAMPLE Simplif the following: ) mn n n This cn e simplified, ut we must e ver creful, ecuse there re two terms in the numertor nd single term in the denomintor. Rewrite, putting ech term of the numertor over the common denomintor: mn n n n simplif dividing out the n in ech frction to otin: mn ) 8 rewrite, putting ech term over the common denomintor. 8 In generl, n n n n 0 0 Module A Introduction to Alger: Prt II
23 EXAMPLE ) rewrite the frction s seprte frctions to get: simplif dividing out the in ech frction to otin: ) rewrite simplif dividing out the 7 in ech frction to otin: c) 0 rewrite the frction s seprte frctions to get: 0 simplif dividing out the in ech frction to otin: Module A Introduction to Alger: Prt II
24 Eperientil Activit Four Simplif the following:.. c c c 6 c p q 9 p q. 7 p q c 6 c c c c 9 c 6.. 8rs t m 8r st rst 6rst n 0mn 7 mn m n w 0 w 8. w 7 8 t t 0. 6 t Show Me. Eperientil Activit Four Answers c c c. (TLM c c c ). (TLM ) pq 7 p 6. (TLM 7p pq ) 6 c 8. c st 9r t 8t (TLM 9r t st 8t) 6 m 6 7mn (TLM 7mn 6 m 6) 6 w w 0. t t (TLM t t ) Note: The nswers for questions,,,, nd 0 hd to e rerrnged to e in the correct TLM formt. The other nswers re were lred in the correct formt. Module A Introduction to Alger: Prt II
25 OBJECTIVE FIVE When ou complete this ojective ou will e le to Perform lgeric long division. Eplortion Activit EXAMPLE ) Perform the division: NOTE: When the denomintor consists of more thn one term, in this cse ( ), we CANNOT simplif dividing out common fctors. Therefore, we ppl the rules of long division to simplif this epression. Rewrite the originl question in the following form: Here is the divisor nd is the dividend. Now in n long division of this stle there re sic steps:. Divide the first term of the divisor into the first term of the dividend nd put wht ou get up top.. Multipl the whole divisor the term ou got in step, nd put the product under the dividend.. Sutrct wht ou got in step from the dividend nd write tht nswer down.. Repet steps to strting dividing the first term of the divisor into the first term of our nswer in step nd continue s mn times s ou hve to until ou get reminder tht is not divisile the first term of the divisor! Check out the following procedure: In the division the FIRST step is to divide into to get. (Step ) multipl this to get : Module A Introduction to Alger: Prt II
26 Sutrct to get 6, nd strt gin dividing into 6 to get, then multipl this the divisor to get 6, then sutrct to get reminder of 0. You re now finished. Thus: Note: rememer ou re sutrcting here so: ( ) Therefore: Check: ( )( ) Notice in this emple the ke words: divisor, dividend, divide, multipl, nd sutrct. These words stte the essence of ever long division prolem. EXAMPLE Perform the division: 7 Rewrite, onl this time leve spces for the missing nd terms nd lso e sure the terms re in descending powers of. Note in this emple the nd the terms re "missing" ecuse their coefficients re 0, i.e. the cn e written in s 0 nd Tret 0 nd 0 s "ordinr" terms. Module A Introduction to Alger: Prt II
27 Module A Introduction to Alger: Prt II The 8 t the end is the reminder. Thus we s ( ) goes into ( 7), [ ] nd 8 times. Compre this to sing goes into 9, 6 nd / times, the red the sme w. So just s 6 9 then 8 7. In most prcticl prolems involving long division the reminder will e something other thn zero. Also, the reminder is lws simpler thn the divisor, i.e. its degree is less thn the degree of the divisor. Thus, when term is reched which is of smller degree thn tht of the divisor, the division is finished nd this term is the reminder. The net emple illustrtes this. EXAMPLE Divide: 0 9 The divisor is not of the st degree. Agin proceed s usul or dd in 0 which fills in the terms nd simplifies the question. The reminder is st degree, therefore simpler thn nd degree divisor. The division is finished!. So 9 9
28 Eperientil Activit Five Perform the indicted division t.. t t 7 t i i M M M 7.. i M Show Me ( ) ( ) ( ) ( ) Eperientil Activit Five Answers.... 9, r., r 6., r , r 6 8., r 0 9., r , r 79., r. t 7t, r. i i 8, r 9. M M 6, r., r , r , r Module A Introduction to Alger: Prt II
29 Prcticl Appliction Activit Complete the Introduction to Alger Prt ssignment in TLM. Summr This module continued with the introductor concepts of eginning lger course. Module A Introduction to Alger: Prt II 7
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