notation, simplifying expressions

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1 I I. Algebr: exponents, scientific nottion, simplifying expressions For more prctice problems nd detiled written explntions, see the following books, both on reserve ll yer in the Sciences Librry. (cn be purchsed.al.g,e.brncl1rigonometry, by Loren C. Lrson in the bookstore) by Keedy nd Bittinger A. Exponents: Definitions nd rules. 1. Definition l =, = -, 3 = --, n = -..- (n times). mn = m+n An exmple showing why: 3 m. - n = m-n o 4..I&f = 1 5. Drl (provided $ 0 ) -n 1 =~ ( $ 0) ( 3 = (-)(--) = + 3 = s [ ) (provided $ C) _ 3 = = = (reson: ~ = 1 nd = = ) o 1 O-n-n (reson: - = L = = ) n n 6. {b)n = n bn ( An exmple showing why: (b)3 = bbb = bbb = 3b3 ] 8. Note: ( + b)n ;II!: n + bn!!! (except when n = 1 or = 0 or b = 0) 1.1

2 e.g. (1+)3= 33= 7,while 13+3=1+8=9. Correct rules: ( + b) = + b + b, ( +b)3 = 3 + 3b + 3b + b3, etc.; see Section D. 9. Exponentition precedes multipliction'. For exmple, 73 = 7..., nq1 (7)3, which would be = (-)n = (-l)nn = { n -n if n even if n odd e.g. (-x) = (-x)(-x) = (_)(_)x =x (-x)3 = (-x)(-x)(-x) = (-)(_)(_)x3 = -x3 Note: -x = (-)x.x ~ (-x). Exmp~ - 1 = x 3-3.' x 5-1 = I (x ) = x ; = = ' xx =L= x ' ' x x 3x4. 5x3 = (3.5)(x4.x3) = 15x7 ; 5-1 or x y Exercises I A 1. x3. 3x 3. (-4x-1z-) (3 -lb -c4)3 ( -lb c-3) Simplify ech expression.. -(x3)4 4. (5x4y-3z)(-xy4z-1) x y. -1 x Y x y x y 1.

3 I B. R~gkgl Jor roots) ncurctionl exponents 1. Definition n.f denotes \ the positive nth root of if n is even nd ) 0 the nth root of if n is odd When is negtive nd n is even (e.g../-9 ), n.; is undefined within the rel number system. ~:.; is short for.f, the positive squre root of. Exmples: ff = ff = 3; 3ff = ; 3./-8 = -;./-9 is undefined.. Definition 1/n is defined to be n.; (where possible) (1..n) 1/n n n 1 1/n. (Reson: ( ) = = =, so ~ the nth root of ) 3. Definition m/n = (1/n)m.QI: (m)1/n (these hve the sme vlue). provided 1/n is defined. 4. n.;b = n.;. njd nd nj71) = n.;/ njd provided n.r- nd njd re both defined. (Note./(-)(-8) =./I6 = 4 even though j: nd ;: re undefined.) 5. n.;+b ~ n.; + njd. In prticulr,./ + b ~ + b, e.g../3 + 4 =./9 +16 = /5 = 5, while 3 +4 = jx = IxI, the bsolute vlue of x. Exmp~ 1. 84/3 = (3jS)4 = 4 = /3 = 3'/18""" = 3'/(3.37)(3.3.) = 3.3'/.3." = 3.3/1 R 14 J? 3. 1L = :i..x:..= 1L but 1L = :i.. 1 = ill J6 = 6;./(-6) = 6; j1 = j4:3 = /4n-= n- 5. Simplify 3fiTP-. 1.3

4 3j7'1)9- - (77b9)1/3. (7)1/37/3b9/3 :I 81/391/37/3b3 =.91/37/3b3 or b3.3./9 (either of these is OK) 6. Rtionlize the denomintor of 13 (eliminte the rdicl). l = 5.13 = sj3 or ~ Simplify (0.07)1/3. (0.07)1/3 = ( 7 )1/3 = ( 333 )1/3 = (33)~/3 = i (103 )1/3 10 or 0.3 Exercises L Simplify: /. (0.008)1/3 4. (x/3)(xl/3)4 3. [(-3)()]1/4 5. x-3/ / x3/ 6. J-5 (~-64,. Express the following using rtionl ( x1/3 v 3/4 ) 7. /3 1/ (x Y ) exponents: 8. ~ x3' 9. JJ; 10. x~ Rtionlize the denomintor: J5 c. Scientific nottion Scientific nottion is uniform wy of writing numbers in which ech number is written in the form k times 10n with 1:s k <10 nd n n integer. Exmp~ 5 = 5 x 100; 5 =.5 x 101 ; 93,000,000 = 9.3 x 107; = 3. x 10-4 ; ,000,000)( ) =.!j..7 x +010)(9.57 x ~ (0.003)(8,000) (3.0 x 10-3)(8. x 104) 10-6 = x = x = x 103 = 6.69 x

5 j Exercises ~. Convert to scientific nottion: 1. 58,000, (30,000)(,700,000). (0.0001)(0.081) Convert to deciml nottion: x x D. prentheses nd mult~ First perform the opertions within the prentheses nd simplify within prentheses where possible. Then crry out the multipliction indicted by the prentheses. (Cution! Be especilly creful with minus signs t this stged Finlly, combine like terms by dding their coefficients. Exmp~ 1. ~3(x - x + 1) = -3x + (-3)(-x) + (-3)(1) = -3x + 3x - 3. (x + 4x + 1) - 3(3x - x + 1-7x + ) = (x + 4x + 1) - 3(-4x - x + 3) = x + 8x + + 1x + 6x + (-9) = 14x + 14x (x + 3(y + 4» + (- + 7(x - y» = -(x + 3y + 1) + (- + 7x - 7y) = -x - 6y x - 14y = 1x - 0y - 8 (work from the inside out) 4. Expnd (3x + 7)(-5x + ). Use the "FOIL" method: Firsts (3x)( -5x) = -15x + Outers (3x)() = 6x + Inners (7)(-5x) = -35x + Lsts (7)() = 14-15x - 9x + 14 F..., 0 ~, (3x + 7)( -5x + ) ~L. I 1.5

6 For more thn two fctors, expnd two t time. Note certin common rules: ( +b)( -b) = - b ( +b)3 =3 + 3b + 3b + b3 (+b) = + b + b (-b)3 = 3-3b + 3b - b3 (-b) = - b + b Exercises I D Perform the indicted opertions, then simplify. 1. (x + y + 7) - (7x - y + 1). (x + 7x + 5)(x) 3. _(x + x(x+1) + 3) + (-5 + (x+1) - 3x) 4. 3[./T-./ /T- 5/3)] 5. (3x + 1)(7x - ) - (3x + 4)(6x - 1) - x(x - 19) 6. (x+ )3 7. (x + 7y) E. Arllhmetic of frctions 1. Addition To dd frctions with the sme denomintor, dd their numertors nd retin the originl denomintor. To dd frctions with unlike denomintors, first find the "lowest common denomintor", then rewrite ech frction so ll hve the "LCD"s denomintor, then dd. Simplify the result if possible by cncelling ny fctors common to numertor nd denomintor..exmp~ 1. ~ = = -8 = i +.. = = 1+14 = , 3 : LCD = 7. 3 = ,3,9=3,6=.3: LCD= = 16 = = =

7 I 4 4x+1 + -L = Hx+:t.)(x+U + 3x = 4x + 4x + x x '. + x x 1 x (x+ 1) (x + 1) x x (x + 1) 7x + 5x + 1 = x (x + 1) or 7x + 5x x + x. ~ Multiply numertors, multiply denomintors, nd simplify if possible. Do ny possible cncelltion of common fctors before. ctully performing the multiplictions. Exmples 1. ~.~ =.3 =.3 = -1- = x+3.-1l- =~ y-4 y+5 (y-4)(y+5) 3. Division reciprocl. or 4 3 x +3x y +y-0 Dividing by frction is the sme s multiplying by its Exmp~ x- / x+5 = x-. x-3 = (x-)(x-31 x+1 x-3 x+1 x+5 (x+1)(x+5) Note: b 1 C b c bc - =-e-=- 4. CompouncUrctions (frctions within frctions) Method 1: Add frctions in numertor. Add frctions in denomintor. Then divide s bove. Method : Multiply numertor nd denomintor by their LCD...Exmp~ Method 1 : x+1. 5 x x = 3x x+1 5 3x-1 3 = 5x+1. ~ 5 3x-l = 15x x - 5 Method : x+1. 5 x (x + 1.)(15) 5 1 (x - - )(15) 3 = 15x x

8 Exercises I E Perform the indicted opertions b 1. (- - -) + (- - - ) c d 1 3. x xy Y A- +-1L -b b- 5. ( )(4+3) (.. + 4x) / (..- x) _ b b_b (x + y ) x + (x) Answers to Exercises I A: 1. 6x5 3. xz4/16 5 ( ) /.1 = x1 4~ -10x6yz 4 x or x y y 7. 7c18 4b or 7-1b 4 c 8. ~ 5 or 8 5 xy B: undefined 5. x-3 or 1/x3 x/3 y3/ /3 x Y x 9. (x1/)1/ = x1/4 11. ff/ or ~ff c: x or /10 or 1/5 4. x/3x4/3 = x 6. ;:5(-4) = /160 = 10 3/4 or x-4/3 8. x x x1x1/3 = x4/

9 (3 x +0 )(.7 x ~ = 8.1 x 10 = 1 x (1 x 10 )(8.1 x 10 ) 8.1 x ,800, x 10 = 3 x 10-4 = x 103 D: 1. -6x + 4y + 6. x3 + 14x + 10x 3. -9x + x ff ~ 5. x - x - 6. x3 + 6x + 1x x + 8xy + 49y E: 1.1. bd + c 4 cd 3. y + x - 3x b 4. 1 (becuse b- - -b x y x 0-3x x y = 1.Y.:!:Z xy =~ y+x 7 b-. b - b 9. ~ =... x x x - --L) 1.9