{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 257 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 17 " Exponent Problems" }}{PARA 0 "" 0 "" {TEXT -1 63 "\nThe rules of expon ents, and solving problems with exponents.\n\n" }{TEXT 258 153 "[Direc tions : Execute the Code Resource section first. Although there will b e no output immediately, these definitions are used later in this work sheet.]" }{TEXT -1 1 "\n" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Cod e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "txln := `\\n_______________________ _____________________________________________________`:\n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "1. Rules of Exponents" }}{PARA 0 "" 0 " " {TEXT -1 79 "\nHere are some of the main rules of exponents, and the ir application in Maple.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "print(cat(txln,`\\n1. Product Rule`));\na^n * a^m: %= simplify(%); print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "10^5 * 10^2 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x^5 * x^4; " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y^8 * y^5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "(x+4)^10 * (x+4)^8;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "x^4 * y^3; `Note that the bases must be the s ame to use this rule`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d istance^2 * distance^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " print(cat(txln,`\\n2. Quotient Rule`));a^n / a^m: %= simplify(%); prin t(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "10^9 / 10^6;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x^14/x^13;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x^20 / x^23;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "q^100 / q^10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(y + 7)^3 / (y+7)^5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "print(cat(txln,`\\n3. Zero Power is One`)); `a^0` = e xpand(a^0);print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " 13^0; 813^0; (-29)^0; (3/19)^0; (41239)^0; (3.141 59)^0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "print(cat(txln,` \\n4. Power of One is Identity`));`a^1` =simplify(a^1); print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "13^1; 813^1; (-29 )^1; (3/19)^1; (41239)^1; (3.14159)^1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "print(cat(txln,`\\n5. Power of a Power`)) ; (a^n)^m = a^(n*m) ; print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(x^2)^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "(y^10)^5; `Compare with ...`, (y^10)*(y^5),` power of a power \255 p roduct of powers`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "((t - 7)^8 )^3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "print(cat(tx ln,`\\n6. Power of Product`)); (a*b)^m = (a^n)*(b^m); print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(3*x)^4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "(8*y)^10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "print(cat(txln,`\\n7. Power of a Product of Powers`) ); \n (a^n*b^m)^m = (a^(n*p))*(b^(m*p)); \n `(This is a combination of rules 1 and 6.)`; print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(x^3 * y^5) ^ 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "(x^10 * y^20) ^ 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(17 * M^7) ^ 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "(3 * x^10 * y^7) ^ 4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "(a^2 * b^3 * c^4 * d^5 ) ^ 10;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 "print(cat(txln,`\\n8. Power of a Quotient`)); \+ (a/b)^n =(a^n)/(b^n); print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(3/4)^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " (x/2)^5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(a/b)^4;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(3/(x+5))^9;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "print(cat(txln,`\\n9. Power of a Q uotient of Powers`));\n (a^n * b^m)^p = a^(n*p)*b^(m* p); \n `(This is a combination of rules 2 and 8.)`; prin t(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "( x^3 / y^ 5)^2 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "( 1 / pressure^2) ^ 3; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(x^10 / 3) ^ 5;" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "2. Negative Exponents" }}{PARA 0 "" 0 "" {TEXT -1 61 "\nThere are a few more rules which involve negati ve exponents." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "print(cat( txln,`\\n10. Negative Exponent is Reciprocal`)); \n \+ a^(-n) = 1/a^n; print(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "3^(-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " `Note :`, 9*3^(-2), 9*(1/9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x^(-13);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(-2)^(-3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(-2)^(-4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "(1/7)^(-2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 124 "print(cat(txln,`\\n11. Reciprocal of a Recipr ocal is Original`)); \n 1/a^(-n) = a^n; p rint(txln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "1/(4^(-3)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "1/y^(-12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "1/(3*x + 2)^(-5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "`Note : Rules 10 and 11 can both be used \+ on the same problem :`;x^(-4) / y^(-9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "print(cat(txln,`\\n12. Reciprocal of a Fraction is Fl ipped`)); (a/b)^(-n) = (b/a)^n; print(txln);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "(2/131)^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "(x/y)^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(1000/A)^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "( ( x+1) / (x+2) )^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "( ( x^2 + 3*x + 1) / (x^2 + 5*x + 2) )^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(( A/B )^(-1))^(-1);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "3. Exponent Problems - Products (Monomials)" }}{PARA 0 " " 0 "" {TEXT -1 366 "\nWhen multiplying two monomial expressions, numb ers can be multiplied, and the same variables can be multiplied - reme mber that we learned in Rule #1 that we can only multiply variables wi th the same base. For example, with two expressions with numbers, x, a nd y's - you multiply :\n - number by number\n - x ter m by x term\n - y term by y term\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "(3 * x^7* y^2) * (4 * x^4 * y^9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "(5 * x^5 * y^2) * (11 * x^11 * y^(-3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "(13 * x^(-3)* y^(-8)) * ( 4 * x^(-2) * y^(-7));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(1 440 * x^100* y^100) * (1200 * x^100 * y^(-100));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(33 * x^33* y^22) * (4 * x^44 * y^55);" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 61 "4. Exponent Problems - Products ( Using Distributive Property)" }}{PARA 0 "" 0 "" {TEXT -1 89 "\nWhen we multiply a term by a sum or difference we need to use the Distributiv e Property." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`Distributive Property`; a*(b+c): % = expand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 147 " \nWe distribute the term being multiplied to each of the terms inside \+ the parentheses. Then each of those multiplications is a product of mo nomials." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(12*x^10)*(14*x^ 3 - 1): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "( 3*x^2)*(14*x + 21): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(2*x^5)*(4*x^5 - x^2): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(a^5*b^10)*(10*a^10*b^9 - a^8*b^7): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "(20*a^5* b)*(13*a^4*b^9 - b^2): % = expand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "5. Exponent Problems - Quoti ents - The Idea" }}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "6. Exponent Prob lems - Quotients - Using Maple" }}{PARA 0 "" 0 "" {TEXT -1 173 "\nWe c an do these problems in Maple too - quite quickly. Try these by hand, \+ and then check to see if your answer agrees with Maple's answer, or is mathematically equivalent. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(36 * x^4 * y^6 ) / (24 * x^5 * y^5);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "(75 * x^10 * y^3 ) / (24 * x * y^5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(100 * x * y ) / (45 * x^11 * y^9); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(32 * x^14 * y^6 ) / (4 0 * x^5 * y^5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "(10 * x^ (-4) * y^(-6) ) / (12 * x^5 * y^5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "(33 * x^(-5) * y^6 ) / (22 * x^(-5) * y^(-5));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "(15 * x^4 * y^(-6) ) / (18 * x^(-4) * y^5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "((6 * x^ 2 * y^3) / (30 * x^4 * y^5 ) ) ^(-1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "((40*x^2*y^9) / (30*x^(-6)*y^8) )^(-2) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "((27 * x^(-32) * y^19) / (24 * x^(- 46) * y^8) )^(-4) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Waterloo Maple Inc " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 33 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }