8 factors of x. For our second example, let s raise a power to a power:
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1 CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish, nd then e ll generlize ht e oserve to the officil Five Ls of Eponents. I. We strt finding the product of nd : 7 ( )( ) 7 fctors of Notice tht the ses (the s) re the sme, nd it s multipliction prolem. As long s the ses re the sme, nd it s multipliction prolem, it ppers tht e merel need to rite don the 7 se, nd then dd the eponents together to get the eponent of the nser. Tht is,. II. For our second emple, let s rise poer to poer: 8 ( ) ( ) ( )( ) 8 fctors of We pper to hve shortcut t hnd. Simpl multipl the to eponents together nd e re done. So, to rise poer to poer, e cn rite generl rule: ( ). 8 ( )
2 III. No e re to tr rising product to poer; for instnce, ( ) ( )( )( )( )( ) ( )( ) 5 fctors of In generl, hen rising product to poer, n n n rise ech fctor to the poer: ( ). ( ) Note tht the quntit in the prentheses is single term -- there s no dding or sutrcting in the prentheses. In fct, if there re to or more terms in the prentheses, this l of eponents does not ppl. IV. Net e divide poers of the sme se. We ll need to emples for this l of eponents. A. 6 B. 6 6 In generl, hen dividing poers of the sme se, sutrct the eponents, leving the remining fctors on the top if the top eponent is igger, nd on the ottom if the ottom eponent is igger. 6 V. Our lst emple in this section is the process of rising quotient to poer. As usul, e stretch nd squish; then e generlize to l of eponents. In generl, e cn rise quotient to poer rising oth the top nd ottom to the n poer: n n Ch 5 The Five Ls of Eponents
3 SUMMARY OF THE FIVE LAWS OF EXPONENTS Eponent L Emple 6 8 ( ) ( ) n n n For, For, ( ) ( z) z 8 n n n SINGLE-STEP EXAMPLES EXAMPLE : A A A A A The ses re the sme, nd it s multipliction prolem. So e cn simpl rite the se nd dd the eponents. Ch 5 The Five Ls of Eponents
4 B. 9 All the ses re the sme, nd it s multipliction prolem, nd so e simpl dd the eponents. C. 9 ( ) ( ) ( ) It doesn t mtter ht the se is, s long s e re multipling poers of the sme se. EXAMPLE : A. 0 0 ( c ) c B. Rising se to poer, nd then rising tht result to further poer requires simpl tht e multipl the eponents. ( ) Poer to poer to poer? Just multipl ll three eponents. EXAMPLE : A ( ) It s poer of product ( single term). So just rise ech fctor to the 5th poer. B ( c) c Even term ith three fctors cn e rised to the 7th poer rising ech fctor to the 7th poer. Ch 5 The Five Ls of Eponents
5 5 EXAMPLE : A. 7 5 Since 7 > 5, e divide poers of the sme se sutrcting the eponents. B Since the igger eponent is on the ottom, e sutrct 5 from 5 nd leve tht poer of on the ottom. EXAMPLE 5: A. z 7 7 z 7 To rise quotient to poer, just rise oth the top nd ottom to the 7th poer. B. ( ) u ( u ) Just rise top nd ottom to the rd poer. Homeork. Use the Five Ls of Eponents to simplif ech epression: c. d. z z e. ( ) f. (z 8 ) g. (n 0 ) 0 h. ( ) 7 Ch 5 The Five Ls of Eponents
6 6 i. () j. (z) 5 k. (RT) l. (mth) 5 m. 8 n. 9 o. 5 5 p. Q Q q. k r. 99 s. m 0 t. (c). Use the Five Ls of Eponents to simplif ech epression:. 5. u 5 u 7 u c. 0 0 d. z z e. ( ) 5 f. (z 9 ) g. (n 00 ) 0 h. ( ) 9 i. () j. (c) 7 k. (pn) l. (love) m. 0 n. o. 9 9 p. Q Q 00 0 q. r. 999 s. z t. () WHEN NOT TO USE THE FIVE LAWS OF EXPONENTS 5 6 cnnot e simplified. Although the first l of eponents demnds tht the epressions e multiplied -- nd the re -- it lso requires tht the ses e the sme -- nd the ren t. + cnnot e simplified. Even though the ses re the sme, the first l of eponents requires tht the to poers of e multiplied. + cn e simplified, ut not the first l of eponents, since the poers of re not eing multiplied. But the to terms re like terms, hich mens e simpl dd them together to get. Ch 5 The Five Ls of Eponents
7 7 ( + ) does not equl +. You m think tht the third l of eponents, () n n n, might ppl, ut it does not, nd tht s ecuse is single term, heres + consists of to terms. You ll hve to it until Intermedite Alger to lern clever to clculte the rd poer of +. Also, ou m hve lred lerned in this clss tht ( + ) + +, nd so gin, ( + ) n n + n. Homeork. Simplif ech epression:.. c. d. p t p e. + 5 f. 5 g. n + n h. i. ( + ) 55 j. Q + Q k. u 5 6 l. h 6 h m. ( ) n. () o. ( ) p. + 5 q. + r. s t u. () v. ( + ).. +. ( )( ) z. n 6 + n 6 MULTI-STEP EXAMPLES EXAMPLE 6: A. ( )(5 7 ) ()(5) ( )( 7 ) 5 0 B. ( ) ( ) ( ) 6 C. Ch 5 The Five Ls of Eponents
8 ( ) 7 D. 5 5 E. F. 7 7 ( ) ( ) ( ) ( ) 5 5 Homeork. Simplif ech epression:. (5 )( ). (7)(7 5 ) c. (u)(u) d. ( ) e. ( + ) f. 0 g. 5m n h. 7 i. 0 j. p q 0( c) k. c 0 l. m. 5 n. ( ) Ch 5 The Five Ls of Eponents
9 9 ZERO AS AN EXPONENT We ve lred lerned tht nthing to the zero poer is (s long s it s not zero to the zero). We reched this conclusion fter e lerned tht 0, nd figured it might e true for n se. No e tr to verif this; tht is, ht is 0? Consider the epression 0 here e ssume 0. To figure out the mening of 0, e cn use the first l of eponents to clculte Tht is, No isolte the 0, since tht s ht e re tring to find the vlue of. We do this dividing ech side of the eqution : 0, hich implies tht It s legl to divide, since e ve stipulted tht 0. nd e re done: 0, An numer (ecept 0) rised to the zero poer is. EXAMPLE 7: A. ( + z) 0 (n quntit ( 0) to the zero poer is ) B. (c) 0 (n quntit ( 0) to the zero poer is ) C (the eponent is on the onl) D. u 0 u() u (the eponent is on the onl) Ch 5 The Five Ls of Eponents
10 0 E. (87) 0 (the eponent is on the 87) F. 0 (the eponent in on the, not on the minus sign) Homeork 5. Evlute ech epression: c d. ( + ) 0 e f. (8 5) 0 + (0 8) g h (0 ) Simplif ech epression: c. + 0 d. ( + ) 0 e. () 0 f. 0 g. ( ) i j. Q 0 Q 0 k. 0 0 l. 0 h. m 0 m Ch 5 The Five Ls of Eponents
11 Revie Prolems 7 7. Simplif: Simplif: 9. Simplif: 9 c c 0. Simplif: ( + ). Simplif: +. Simplif: u + u. Simplif: cd 0 e 0. Simplif: Solutions.. 7. c. 6 d. z e. f. z 6 g. n 00 h. 7 i. j. 5 5 z 5 k. RT l. m 5 5 t 5 h 5 m. 6 n. o. p. Q 50 6 q. k r s. 0 m t. c.. 8. u c. 60 d. z 5 e. 0 f. z 8 g. n 000 h. 9 i. j. 7 7 c 7 k. pn l. l o v e m. 8 n. 9 o. p. Q 80 Ch 5 The Five Ls of Eponents
12 q. r s. z t As is c. 9 d. p 5 t e. As is f. 8 g. n h. 0 i. As is (for no) j. Q k. As is l. As is m. + n. o. 9 p. As is q. r. 0 s. As is t. 0 u. v As is. As is. z. n c. u d. 5 e f. 8 g. 5m 9 n 0 h. 8p q i j. 80c 9 k. l m. 0 0 n. 6 8 c c. d. e. 5 f. g. 0 h c. + d. e. f. g. h. m i. j. k. 0 l c As is. u. c. 5 5 I hve no prticulr tlent. I m merel inquisitive. Alert Einstein Ch 5 The Five Ls of Eponents
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