Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive. The rst is considered in the following exmple, which is written out in two dierent wys. Exmple. Simplify. Use quotient rule; subtrct exponents 0 ; now consider the problem in the second wy Rewrite exponents; use repeted multipliction Reduce out ll the 0 s ; combine the two solutions; get: 0 Our Finl Solution This nl result is n importnt property known s the zero power rule of exponents. Zero Power Rule of Exponents: 0 Any non-zero number or expression rised to the zero power will lwys be. This is illustrted in the following exmple. Exmple 2. Simplify. (3x 2 ) 0 Zero power rule Here we re ssuming tht x is not 0. If x0, then (3 0 2 ) 0 would equl 0, not. Another property we will consider here dels with negtive exponents. Agin we will solve the following exmple in two wys. 85
Exmple 3. Simplify nd write the nswer using only positive exponents. 5 2 Use the quotient rule; subtrct exponents ; solve this problem nother wy 5 2 2 2 Rewrite exponents s repeted multipliction Reduce three 0 s out of top nd bottom Simplify to exponents ; combine the two solutions; get: Our Finl Solution This exmple illustrtes n importnt property of exponents. Negtive exponents yield the reciprocl of the bse. Once we tke the reciprocl, the exponent is now positive. Also, it is importnt to note tht negtive exponent does not men tht the expression is negtive; only tht we need the reciprocl of the bse. The rules of negtive exponents follow. Rules of Negtive Exponents: m b b m Negtive exponents cn be combined in severl dierent wys. As generl rule, if we think of our expression s frction, negtive exponents in the numertor must be moved to the denomintor; likewise, negtive exponents in the denomintor need to be moved to the numertor. When the bse with exponent moves, the exponent is now positive. This is illustrted in the following exmple. Exmple 4. Simplify nd write the nswer using only positive exponents. b 2 c Move negtive exponents on b; d; nd e; 2d e 4 f 2 exponents become positive cde 4 2b 2 f 2 86
As we simplied our frction, we took specil cre to move the bses tht hd negtive exponent, but the expression itself did not become negtive becuse of those exponents. Also, it is importnt to remember tht exponents only eect the bse they re ttched to. The 2 in the denomintor of the bove exmple does not hve n exponent on it, so it does not move with the d. We now hve the following nine properties of exponents. It is importnt tht we re very fmilir with ll of them. Note tht when simplifying expressions involving exponents, the nl form is usully written using only positive exponents. Properties of Exponents n +n (b) m b m n m n b m b m m ( ) n n 0 b b m World View Note: Nicols Chuquet, the French mthemticin of the 5th century, wrote 2 m to indicte 2x. This ws the rst known use of the negtive exponent. Simplifying with negtive exponents is much the sme s simplifying with positive exponents. It is the dvice of the uthor to keep the negtive exponents until the end of the problem nd then move them round to their correct loction (numertor or denomintor). As we do this, it is importnt to be very creful of rules for dding, subtrcting, nd multiplying with negtives. This is illustrted in the following exmples. Exmple 5. Simplify nd write the nswer using only positive exponents. 4x 5 y 3 3x 3 y 2 6x 5 y 3 2x 2 y 5 6x 5 y 3 2x 3 y 8 2x 3 y 8 Simplify numertor with product rule; dd exponents Use quotient rule; subtrct exponents; be creful with negtives ( 2) ( 5) ( 2) + 5 3 ( 5) 3 ( 5) + ( 3) 8 Move negtive exponent to denomintor; exponent becomes positive 87
Exmple 6. Simplify nd write the nswer using only positive exponents. (3b 3 ) 2 b 3 2 4 b 0 3 2 2 b 6 b 3 2 4 In numertor; use power rule with 2; multiply exponents In denomintor; b 0 In numertor; use product rule; dd exponents 3 2 b 9 2 4 3 2 b 9 2 Use quotient rule; subtrct exponents; be creful with negtives ( ) ( 4) ( ) + 4 3 Move 3 nd b to denomintor becuse of negtive exponents; exponents become positive 3 2 2b 9 Evlute 3 2 2 8b 9 In the previous exmple it is importnt to point out tht when we simplied 3 2, we moved the three to the denomintor nd the exponent becme positive. We did not mke the number negtive! Negtive exponents never mke the bses negtive; they simply men we hve to tke the reciprocl of the bse. One finl exmple with negtive exponents is given here. Exmple 7. Simplify nd write the nswer using only positive exponents. 3x 2 y 5 z 3 6x 6 y 2 z 3 9(x 2 y 2 ) 3 3 In numertor; use product rule; dding exponents In denomintor; use power rule; multiplying exponents 8x 8 y 3 z 0 9x 6 y 6 3 Use quotient rule to subtrct exponents; be creful with negtives: ( 8) ( 6) ( 8) + 6 2 3 6 3 + ( 6) 3 (2x 2 y 3 z 0 ) 3 Prentheses re done; use power rule with 3 2 3 x 6 y 9 z 0 Move 2 with negtive exponent down nd z 0 ; exponent of 2 becomes positive x 6 y 9 Evlute 2 3 2 3 x 6 y 9 8 88
3.2 Prctice Simplify ech expression. Your nswer should contin only positive exponents. ) 2x 4 y 2 (2xy 3 ) 4 2) 2 2 b 3 (2 0 b 4 ) 4 3) ( 4 b 3 ) 3 2 b 2 4) 2x 3 y 2 (2x 3 ) 0 5) (2x 2 y 2 ) 4 x 4 6) (m 0 n 3 2m 3 n 3 ) 0 7) (x 3 y 4 ) 3 x 4 y 4 8) 2m n 3 (2m n 3 ) 4 9) 0) 2x 3 y 2 3x 3 y 3 3x 0 3y 3 3yx 3 2x 4 y 3 ) 4xy 3 x 4 y 0 4y 2) 3) 3x 3 y 2 4y 2 3x 2 y 4 u 2 v 2u 0 v 4 2uv 4) 2xy2 4x 3 y 4 4x 4 y 4 4x 5) u 2 4u 0 v 3 3v 2 6) 2x 2 y 2 4yx 2 7) 2y (x 0 y 2 ) 4 8) (4 ) 4 2b 9) ( 22 b 3 ) 4 20) ( 2y 4 x 2 ) 2 2) 22) 2nm 4 (2m 2 n 2 ) 4 2y 2 (x 4 y 0 ) 4 23) (2mn)4 m 0 n 2 24) 2x 3 (x 4 y 3 ) 25) y3 x 3 y 2 (x 4 y 2 ) 3 26) 2x 2 y 0 2xy 4 (xy 0 ) 27) 2u 2 v 3 (2uv 4 ) 2u 4 v 0 28) 2yx2 x 2 (2x 0 y 4 ) 29) ( 2x0 y 4 y 4 ) 3 30) u 3 v 4 2v(2u 3 v 4 ) 0 3) y(2x4 y 2 ) 2 2x 4 y 0 32) 33) b (2 4 b 0 ) 0 2 3 b 2 2yzx 2 2x 4 y 4 z 2 (zy 2 ) 4 34) 2b4 c 2 (2b 3 c 2 ) 4 2 b 4 35) 2kh0 2h 3 k 0 (2kj 3 ) 2 36) ( (2x 3 y 0 z ) 3 x 3 y 2 2x 3 ) 2 37) (cb3 ) 2 2 3 b 2 ( b 2 c 3 ) 3 89
38) 2q4 m 2 p 2 q 4 (2m 4 p 2 ) 3 39) (yx 4 z 2 ) z 3 x 2 y 3 z 40) 2mpn 3 (m 0 n 4 p 2 ) 3 2n 2 p 0 90
) 32x 8 y 0 2) 32b3 2 3) 25 b 4) 2x 3 y 2 5) 6x 4 y 8 6) 7) y 6 x 5 32 8) m 5 n 5 9) 2 9y 0) y5 2x 7 ) y 2 x 3 2) y8 x 5 4 u 3) 4v 6 4) x7 y 2 2 u 2 5) 2v 5 6) y 2x 4 7) 2 y 7 8) 6 2b 9) 6 2 b 2 20) y8 x 4 4 2) 8m 4 n 7 22) 2x 6 y 2 23) 6n 6 m 4 24) 2x y 3 25) 3.2 Answers x 5 y 26) 4y 4 27) u 2v 28) 4y 5 29) 8 30) 2u 3 v 5 3) 2y 5 x 4 32) 3 2b 3 33) 34) 35) x 2 y z 2 8c 0 b 2 h 3 k j 6 36) x30 z 6 6y 4 37) 2b4 2 c 7 38) m4 q 8 4p 4 39) x 2 y 4 z 4 40) mn7 p 5 9