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Math Help - rationalize expressions

  1. #1
    Newbie
    Joined
    Sep 2007
    Posts
    9

    rationalize expressions

    hi i was wondering if any one can help me rationalize these four expressions and if possible show working out so i can understand better, thanks in advance.

    1)
    8_
    √2

    2)
    16 √9
    √12 3√8


    3
    √2 (1+√2)

    4)
    1+
    √2
    1- √2
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    To rationalize is to make the denominator a rational number. Meaning, do not leave a surd or radical number in the denominator.

    That's my understanding, so:

    8 /sqrt(2)
    Multiply both numerator and denominator by sqrt(2),
    = [8sqrt(2)] /2
    = 4sqrt(2) --------------------answer.

    [16sqrt(9)] /[sqrt(12) *3sqrt(8)]

    There many ways to do this.
    One way is to simplify first all of the radicals.

    = [16*3] /[sqrt(4*3) *3sqrt(4*2)]
    = 48 /[2sqrt(3) *6sqrt(2)]
    = 48 /[12sqrt(3*2)
    = 4 /sqrt(6)
    Multiply both numerator and denominator by sqrt(6),
    = 4sqrt(6) /6
    = (2/3)sqrt(6) ---------------answer.

    sqrt(2)*[1 +sqrt(2)]
    = [sqrt(2) *1 ] +[sqrt(2) *sqrt(2)]
    = sqrt(2) +2 ------------------------------answer.

    [1 +sqrt(2)] /[1 -sqrt(2)]

    That denominator needs to be multiplied by its conjugate [1 +sqrt(2)]---which just happens to be the numerator here---to make the denominator a difference of two squares.

    Multiply both numerator and denominator by [1 +sqrt(2)],
    = [1 +sqrt(2)]^2 /[1 -2]
    = [1 +2sqrt(2) +2] /[-1]
    = -[3 +2sqrt(2)]
    = -3 -2sqrt(2) -------------------answer.
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