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Math Help - Without the use of a calculator

  1. #1
    Super Member dhiab's Avatar
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    Without the use of a calculator

    Without the use of a calculator , calculate :
     <br />
\sqrt{1000\times1001\times1002\times1003}<br />
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  2. #2
    Super Member Bacterius's Avatar
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    What do you mean by "calculating" ? You mean evaluate, because 1000 \times 1001 \times 1002 \times 1003 is definitely not a perfect square (it is nearly one, but it's not) ... so we can only approach it !
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  3. #3
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    Quote Originally Posted by dhiab View Post
    Without the use of a calculator , calculate :
     <br />
\sqrt{1000\times1001\times1002\times1003}<br />
    Define
    z = x^3 \; (x^3 + 1) \; (x^3+2) \; (x^3+3)
    so
    z = [x^3 \; (x^3+3)] \; [(x^3+1) \; (x^3+2)]
    z = (x^6 + 3x^3)(x^6 + 3x^3 +2)

    If we let y = x^6 + 3x^3
    then
    z = y (y+2) = y^2 + 2y = (y+1)^2 -1
    so for large y,
    \sqrt{z} \approx y+1

    Now take x = 10, whence y+1 = 10^6 + 3 \times 10^3 + 1
    Last edited by awkward; November 14th 2009 at 11:30 AM. Reason: typo
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