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Exponents and Radicals : Exercise (2.4) - Solution

  1. Simplify the following.

    (a) 35+75


    35+75=105


    (b) 7512


    7512=25×34×3=5323=33


    (c) 333327


    333327=27327=2739×3=27333=8133=813=243


    (d) 2532


    2532=610


    (e) (43)2


    (43)2=162×4×3+3=1983


    (f) (3+22)(3+2)


    (3+22)(3+2)=3(3+22)+2(3+22)=3+26+6+4=7+36


    (g) (76)(7+6)(x+1)(x1)


    (76)(7+6)(x+1)(x1)=(76)(x1)=x1


    (h) 753448512


    753448512=25×33416×354×3=5334×435×23=5333103=83


    (i) 2x2+532x2298x2


    2x2+532x2298x2=2x2+516×2x2249×2x2=(2x)+(5×42x)(2×72x)=2x+202x142x=72x


    (j) 20a3+a5a+80a3


    20a3+a5a+80a3=4×5a3+a5a+16×5a3=2a5a+a5a+4a5a=7a5a


  2. Rationalise the denominators and simplify.

    (a) 25


    25=25×55=255


    (b) 52+3


    52+3=5(23)(2+3)(23)=5(23)43=5(23)


    (c) 1253


    1253=12(5+3)(53)(5+3)=12(5+3)53=12(5+3)2=6(5+3)


    (d) 2+1221


    2+1221=(2+1)(22+1)(221)(22+1)=2(22+1)+1(22+1)81=4+2+22+17=5+327


    (e) 7+3272


    7+3272=(7+32)(7+2)(72)(7+2)=7(7+2)+32(7+2)72=7+14+314+65=13+4145


    (f) 171117+11


    171117+11=(1711)(1711)(17+11)(1711)=1721711+111711=2821876=2(14187)6=141873


    (g) 1223


    1223=1(22+3)(223)(22+3)=22+383=22+35


    (h) 6+135


    6+135=(6+1)(3+5)(35)(3+5)=6(3+5)+1(3+5)95=36+30+3+54=3+5+36+304


  3. Write as a single fraction.

    (a) 13+1+131


    13+1+131=13+1×3131+131×3+13+1=31(3)212+3+1(3)212=3131+3+131=31+3+12=232=3


    (b) 27+2+172


    27+2+172=27+2×7272+172×7+27+2=2(72)(7)2(2)2+7+2(7)2(2)2=2(72)72+7+272=2722+7+25=3725


    (c) 13+3+133+13


    13+3+133+13=3333(3+3)(33)+33+33(3+3)(33)+63(3+3)(33)=333+33+363(3+3)(33)=03(3+3)(33)=0


    (d) 7+575+11311+3


    7+575+11311+3=(7+5)(7+5)(75)(7+5)+(113)(113)(11+3)(113)=54+145495+20611119=54+14544+206112=27+7522+10311=27+75+220661122=247+75661122


    (e) 3+22(31)2


    3+22(31)2=3+22(3)223+1=3+22423×4+234+23=(3+22)(4+23)42(23)2=12+63+82+461612=12+63+82+464=6+33+42+262


    (f) x+1x1+x1x+11x21


    x+1x1+x1x+11x21=x+1x1+x1x+11x21=(x+1x1×x1x1)+(x1x+1×x+1x+1)(1x21×x21x21)=x21x1+x21x+1x21x21=(x+1)x21(x1)(x+1)+(x1)x21(x+1)(x1)x21x21=(x+1+x11x21)x21=(2x+1)x21x21


    (g) 532+42223


    532+42223=525+4122123=2+21418=41/8=4×8=32=42


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