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Math Help - Quickie #15

  1. #1
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    Quickie #15


    Simplify: . \left(1 + 2^{-\frac{1}{32}}\right)\left(1 + 2^{-\frac{1}{16}}\right)\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)


    Edit: 34 views and no soltuion yet?
    Last edited by Soroban; January 24th 2007 at 05:07 PM.
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  2. #2
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    Finally found one that stumped all of you?


    Let: . X \;=\;\left(1 + 2^{-\frac{1}{32}}\right)\left(1 + 2^{-\frac{1}{16}}\right)\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)


    Multiply both sides by \left(1 - 2^{-\frac{1}{32}}\right)

    \left(1 - 2^{-\frac{1}{32}}\right)X \;=\;\underbrace{\left(1 - 2^{-\frac{1}{32}}\right)\left(1 + 2^{-\frac{1}{32}}\right)}\left(1 + 2^{-\frac{1}{16}}\right) \left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)

    . . . . . . . . . . . . . . = \;\underbrace{\left(1 - 2^{-\frac{1}{16}}\right) \left(1 + 2^{-\frac{1}{16}}\right)}\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)

    . . . . . . . . . . . . . . . . . . = \;\underbrace{\left(1 - 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{8}}\right)}\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)

    . . . . . . . . . . . . . . . . . . . . . . =\;\underbrace{\left(1 - 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{4}}\right)}\left(1 + 2^{-\frac{1}{2}}\right)

    . . . . . . . . . . . . . . . . . . . . . . . . . . = \;\underbrace{\left(1 - 2^{-\frac{1}{2}}\right)\left(1 + 2^{-\frac{1}{2}}\right)}

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = \;1 - 2^{-1}

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =\;\frac{1}{2}


    Therefore: . X \;= \;\frac{\frac{1}{2}}{1 - 2^{-\frac{1}{32}}} \;=\;\frac{1}{2 - 2^{\frac{31}{32}}}<br /> <br />

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  3. #3
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    Thankyou! I was waiting forever for someone to do this!
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    Quote Originally Posted by Soroban View Post
    Finally found one that stumped all of you?
    Of course not! My calculator came up with 23.334023182557 a long time ago!

    -Dan
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