{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 12 " Square Roots" }}{PARA 0 "" 0 "" {TEXT -1 207 "\nWe examine the notion \+ of the square root graphically and numerically, then use this understa nding to simplify square roots and square root expressions. We also lo ok at fractional exponents for square roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Cod e Resource section first. Although there will be no output immediately , these definitions are used later in this worksheet.]" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "restart; with(plots): with(RealDomain):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 563 "StandardPlot := proc( number )\n local N, n, A, h, v, i, ps;\nN := floor(number):\nn := ceil( sqrt(N) ): \nA := plot( sqrt(x), x = -1..(N+1), tickmarks = [N-1,n-1] ):\nh := seq(plot( [[0,k],[N,k]], color = blue), k = 1..n):\nv := seq(plot( [[ k,0],[k,n]], color = blue), k = 1..N):\nfor i from 0 to N do\n if( floor( sqrt(i)) = sqrt(i) ) \n then ps||i := plottools[disk]([i, s qrt(i)], .25, color=red):\n else ps||i := plottools[disk]([i, sqrt( i)], .15, color=yellow):\n fi:\nod:\n\nplots[display]( \{ A,h,v,seq (ps||i, i = 0..N)\}, scaling=constrained );\nend proc:\n\n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 "1. Examing A Graph of the Square Root" }}{PARA 0 "" 0 "" {TEXT -1 113 "If we look at the graph the square roo t on a grid, we see that the curve only passes through certain grid po ints." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "StandardPlot(10);\n " }}}{PARA 0 "" 0 "" {TEXT -1 117 "In particular, the curve hits the g rid intersections at x = 0, 1, 4, 9. Let's look at a large diagram of \+ this sort.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "StandardPl ot(27);\n" }}}{PARA 0 "" 0 "" {TEXT -1 136 "We see red dots at x = 0,1 , 4, 9, 16, 25. It appears that the square root has integer values onl y for values \nwhich are perfect squares." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "2. Tables" }}{PARA 0 "" 0 "" {TEXT -1 228 "\nHere is a table of perfect squares. The left column is a number, and the right number its square. You may recognize the n umbers in the right column as the same numbers which the square root c urve crossed the grid intersections." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "array( [seq( [ k, k^2 ],k = 1..18)]);" }}}{PARA 0 "" 0 "" {TEXT -1 107 "\n\nWhat number, when squared, gives 121? The answe r to this is the \"square root\" - the inverse of the square" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(121);" }}}{PARA 0 "" 0 "" {TEXT -1 80 "And sure enough, if we check it by squaring, we get ba ck to the original number." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "11^2;" }}}{PARA 0 "" 0 "" {TEXT -1 38 "\nWhat number, when squared, g ives 324?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sqrt(324); " }} }{PARA 0 "" 0 "" {TEXT -1 38 "What number, when squared, gives 2500?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sqrt(2500); " }}}{PARA 0 " " 0 "" {TEXT -1 170 "\nIf we write these numbers in the other directio n, we have a square root table. This shows that the square root of a n umber which is a square is just the original number." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "array( [seq( [ k^2 , k],k = 1..13)]);" }} }{PARA 0 "" 0 "" {TEXT -1 165 "\nWhat about the square roots of other \+ numbers, which are not perfect squares? A square root of a whole numbe r is either another whole number or an irrational number." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "array( [seq( [ k , sqrt(k) = evalf( sqrt(k), 15) ],k = 1..13) ]);\n" }}}{PARA 0 "" 0 "" {TEXT -1 100 "\n For this reason, it's a good idea to memorize some perfect squares, so you'll recognize them later." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "array( [seq( [ k , k^2],k = 1..20) ]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "array( [seq( [ 5*k , (5*k)^2],k = 1..20) ]);" } }}{PARA 0 "" 0 "" {TEXT -1 144 "\nAnd when the number is NOT a perfect square root you can leave it in radical form, or use Maple to find pa rt of the infinite decimal expansion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sqrt( 101): % = evalf(%,20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sqrt( 214): % = evalf(%,20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sqrt( 1333): % = evalf(%,20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sqrt( 123349232401 ): % = evalf(%,2 0);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "3. Simplifying Square Ro ots" }}{PARA 0 "" 0 "" {TEXT -1 83 " \nWhen a number is a perfect squa re, its square root is simply the original number." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "sqrt(16); sqrt(51^2);" }}}{PARA 0 "" 0 "" {TEXT -1 129 "\nThis same principle also applies to variables. The squ are root of a perfect square is the absolute value of the original num ber." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sqrt( x^2): % = simp lify( %);" }}}{PARA 0 "" 0 "" {TEXT -1 101 "\nWhen a number is not a p erfect square, it may be square free, in which case there is nothing \+ to do." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sqrt( 3*5*11);\n\n " }}}{PARA 0 "" 0 "" {TEXT -1 73 "\nOr there may be embedded perfect s quares - which are not always obvious." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sqrt( 3^3 ); sqrt( 27 );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "sqrt( 2*25); sqrt( 50 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sqrt( (2^5)*(3^4)*(5^3)*7);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "4. Fractional \+ Exponents." }}{PARA 0 "" 0 "" {TEXT -1 70 "\nNotice that Maple represe nts number to the 1/2 power as square roots." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "3^(1/2); " }}}{PARA 0 "" 0 "" {TEXT -1 130 "\nThi s is because these are the same thing. Note if we square both a square root, or a fractional exponent, we get the same result." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sqrt( a )^2; \n(a^(1/2))^2;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 259 35 " \+ \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }