Objective: To simplify quotients using the Laws of Exponents. Laws of Exponents. Simplify. Write the answer without negative exponents. 1.
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1 Qotients of Monomils Objetive: To simplif qotients sing the Lws of Eponents. Lws of Eponents m n = m n ( b ) m = m b m ( m ) n = m n n m n m = m n n m = m m m b b = Prtie Problems Simplif. Write the nswer withot negtive eponents
2 . p p. ( ) v v 5. ( ) ( ) ( ) 6 6
3 6. ( ) ( ) ( ) m n m n 7. ( ) ( ) n n n n t t t t t
4 Zero nd Negtive Eponents Objetive: To simplif epressions involving the eponent zero nd negtive integrl eponents. If n is positive integer nd 0: 0 = n = n Prtie Problems Write in simplest form withot negtive eponents.. 6 m n n m n m. ( ) 0 5 6
5 r t r r t t r t r 5. ( ) ( )
6 Rtionl Epressions Objetive: To simplif rtionl lgebri epressions. A rtionl nmber is one tht n be epressed s qotient of integers. Similrl, rtionl epression, is one tht n be epressed s qotient of polnomils. A rtionl epression is simplified when it is epressed s qotient of polnomils whose gretest ommon ftor is (i.e., the polnomils re reltivel prime. For emple, 5 nd 6 re prime reltive to eh other. Therefore, the re reltivel prime.) Prtie Problems Simplif ( 5 6 ) ( 9 ) ( ). ( 5 ) ( 6 5 )
7 . ( )( ) ( ) w 8 0 w 90 w w 6 w w 5 0 w 5
8 Find the ) domin, nd b) zeros of eh fntion.... to do this, ftor both nmertor nd denomintor. DO NOT nel ot n terms tht m be ommon to both nmertor nd denomintor. to determine the domin, find the nmbers tht mke the denomintor zero. These nmbers re elded from the domin. to determine the zeros, find the nmbers tht mke the nmertor zero. Mke sre these nmbers re in the domin (i.e., re not elded). to simplif, nel ot n ommon terms. YOU MUST DO THIS LAST. ( 0 ). f ( ) = 6. h ( ) = ( )( 9 ) ( )( )
9 . r ( ) =. f ( ) = 8
10 Prodts nd Qotients of Rtionl Epressions Objetive: To mltipl nd divide rtionl epressions Prodts of rtionl epressions shold lws be epressed in simplest form. Prtie Problems Simplif
11 . ( ) ( )
12 Sms nd Differenes of Rtionl Epressions Objetive: To dd nd sbtrt rtionl epressions... when dding (or sbtrting) rtionl epressions, the first thing o hve to do is find the lest ommon denomintor. Then o jst dd, or sbtrt, nd simplif the frtion. Prtie Problems Simplif d d ( d ) d d
13 . 5. ( ) 6. Find onstnts A nd B tht mke the eqtion tre = B A
14 Comple Frtions Objetive: To simplif omple frtions To simplif omple frtions: Simplif the nmertor nd denomintor. Simplif eh seprtel. Divide. Prtie Problems Simplif ( ) ( ) t r t r t r t r t r t r t r
15 ( )( ) 6. ( ) ( ) 7. ( ) ( ) ( ) ( ) k k k k k k k k k
16 8. ( ) ( ) ( ) ( ) ( ) ( ) 9. If f ( ) = ) f ( h ) f ( ) h, epress, nd b) [ f ( f ( ) ) ( 6 ) ] s single simplified frtions.
17 Prtie Problems. Mltiple Choie.. ( ) ) b) ) d). If ( ) f =, simplif ( ) ( ) f f ) b) ) d). ( ) ( ) ) b) ) d)
18 Frtionl Coeffiients & Eqtions Objetive: To solve eqtions nd ineqlities hving frtionl oeffiients. To solve n eqtion or n ineqlit hving frtionl oeffiients, o hve to mltipl both sides of the open sentene b the lest ommon denomintor. Prtie Problems Solve. 5. b b is of 5 of nmber. Find the nmber.. 75% of 60% of the nmber is 6. Wht is the nmber?
19 5. An old onveor belt tkes hors to move one d s ol otpt from the mine to ril line. A new belt n do it in 5 hors. How long does it tke when both re sed t the sme time? 6. How mh wter mst be evported from 00 L tnk of % slt soltion to obtin 5% soltion? % slt Amont Amont soltion slt
20 7. Pm jogged p hill t 6 km/h nd then jogged bk down t 0 km/h. How mn kilometers did she trvel in ll if her totl jogging time ws hor nd 0 mintes? Uphill 6 t rte * time = distne Downhill 0 t 8. Lin invested $,000, prt t 8% nd the rest t 7.%. How mh did she invest t eh rte if her inome from the 8% investment is two thirds tht of the 7.% investment?
21 9. A phrmist wishes to mke.8 L of 0% soltion of bori id b miing 7.5% nd % soltions. How mh of eh tpe of soltion shold be sed? % id Amont Amont soltion id 0. How mh of n 8% soltion of slfri id shold be dded to 60 ml of 0% soltion to obtin 5% soltion? % id Amont Amont soltion id
22 . The ont s new pving mhine n srfe km of highw in 0 h. A mh older mhine n srfe km in 8 h. How long will it tke them to srfe km of highw if the strt t opposite ends nd work d nd night?. Pipes A nd B n fill storge tnk in 8 h nd h, respetivel. With the tnk empt, pipe A ws trned on t noon, nd then pipe B ws trned on t :0 P.M. At wht time ws the tnk fll? Pipe A: Pipe B: job 8 hors job hors Pipe A works lone for hors: job , this mens tht Pipe A fills 6 So, together, the two pipes hve to fill... o finish it... of tnk. 6 of tnk.
23 . A r rditor is filled with 5 L of 5% ntifreeze soltion. How mn liters mst be drwn off nd repled b 75% ntifreeze soltion to leve the rditor filled with 55% ntifreeze soltion? % ntifreeze Amont Amont soltion ntifreeze
24 An eqtion in whih vrible ors in denomintor, sh s = is lled frtionl eqtion. Agin, o hve to mltipl both sides b the LCD of ll the frtions. The reslting eqtion is not lws eqivlent to the originl one nd this n reslt in etrneos roots. These etrneos roots re not roots of the originl eqtion nd do not form prt of the soltion. Prtie Problems Solve.. 9 = 5. = 0. =
25 . Find two nmbers whose sm is 5 nd the sm of whose reiprols is The reiprol of one third of nmber deresed b one third of the reiprol of the nmber is. Find the nmber.
26 6. Helped b strong jet strem, Los Angeles-to- Boston plne flew 0% fster thn sl nd mde the 00 km trip in 0 min less time thn sl. At wht speed does the plne sll fl? rte * time = distne ginst jet strem with jet strem r t 00. r t.5 00 So, this gives s two eqtions: r * t = 00 nd. r ( t.5 ) = 00 t = 00 r. r ( 00 r.5 ) = 00. ( r ) = r = 00 0 =.605 r r =
27 7. To mesre the speed of the jet strem, wether plne left its bse t noon nd flew 800 km diretl ginst the strem with n ir speed of 750 km/h. It then retrned diretl to its bse, rriving t : P.M. Wht ws the speed of the jet strem? rte * time = distne ginst jet strem with jet strem 750 w t w t 800 Totl trvel time: hor nd mintes hors 5 t t = w 750 w = 5
28 8. Bese of trffi Mri old verge onl 0 km/h for the first 0% of her trip, bt she verged 75 km/h for the whole trip. Wht ws her verge speed for the lst 80% of her trip? rte * time = distne st Prt (0%) 0. d nd Prt (80%) r.8 d Totl time: t = d 75
29 9. Pipe A n fill tnk in 5 hors. Pipe B n fill it in hors less time thn it tkes pipe C, drinpipe, to empt the tnk. With ll three pipes open, it tkes hors to fill the tnk. How long wold it tke pipe C to empt it? Pipe A: Pipe C: Pipe B: job 5 hors job hors job hors fills drins fills With ll three pipes open: = 5 ( ) 5 5 ( ) = 5 ( ) = ( ) 5 0 = ( )
30 ( 0 ) = = = 0 0 = 5 0 = ( 5 ) ( ) = 5, it will tke Pipe C 5 hors to empt the tnk.
31 ... more word problems.... I rn into this g the other d tht ws selling rell inepensive 0-pond bgs of ement mi. Well, it trns ot tht the mi ws 5% ement nd 75% snd. S I boght one of these bgs. How mh pre ement mst I dd to get 0% ement mi? % ement Amont mi Amont ement 5% 0.5 ( 0 ) 00% 0% 0. ( 0 )
32 . A 5 horsepower (hp) pmp n empt pool in 5 hors. A smller, hp pmp empties the sme pool in 8 hors. The pmps re sed together to begin empting this pool. After two hors, the hp pmp breks down. How long will it tke the lrger pmp to empt the pool? 5 8 = this is how mh of the pool ws emptied with both pmps working together. Now, with the bigger pmp working lone, 5 t = = t 5... o know, I m rell tired of tping this stff ( better word omes to mind, bt let s jst ll it stff ); so, o finish it.
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