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Math Help - Please help with a short cut solution. There's got to be one I know!!!

  1. #1
    cogscineuro
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    Exclamation Please help with a short cut solution. There's got to be one I know!!!

    (187)273 X (456)135 X (299)424 X (588)139 X (394)154 X (181)177 X
    (1342)437 = .....x

    This is a sum from Indices.
    That would be 187 raised to the power 273 and likewise for the other
    numbers... the superscript didn't show up.

    There's got to be a shortcut method to solve this. I just can't see
    it. Pleeeeeeeeeease Help!!! I need this at the earliest by thursday.

    Thankyou sooo much.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by cogscineuro View Post
    (187)273 X (456)135 X (299)424 X (588)139 X (394)154 X (181)177 X
    (1342)437 = .....x

    This is a sum from Indices.
    That would be 187 raised to the power 273 and likewise for the other
    numbers... the superscript didn't show up.

    There's got to be a shortcut method to solve this. I just can't see
    it. Pleeeeeeeeeease Help!!! I need this at the earliest by thursday.

    Thankyou sooo much.
    Logs! (in what follows log denotes the common log, that is base 10)

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437) =
    .........273 log(187) + 135 log(456) + 424 log(299) + 139 log(588) + 154 log(394) + 177 log(181) + 437 log(1342)~=4579.9444

    So the product ~=8.799 x 10^4579

    RonL
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Logs! (in what follows log denotes the common log, that is base 10)

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437) =
    .........273 log(187) + 135 log(456) + 424 log(299) + 139 log(588) + 154 log(394) + 177 log(181) + 437 log(1342)~=4579.9444

    So the product ~=8.799 x 10^4579

    RonL
    This, by the way, is how your calculator does the problem.

    -Dan
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by topsquark View Post
    This, by the way, is how your calculator does the problem.

    -Dan
    My calculator (if I could find where the children have put any of them)
    wouldn't, it/they cant display a number with a four digit decimal exponent!

    RonL
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    Logs! (in what follows log denotes the common log, that is base 10)

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437) =
    .........273 log(187) + 135 log(456) + 424 log(299) + 139 log(588) + 154 log(394) + 177 log(181) + 437 log(1342)~=4579.9444

    So the product ~=8.799 x 10^4579

    RonL
    In order to understand what is going on here you need to know
    some of the laws of logarithms.

    1. log(A*B)=log(A) + log(B)

    2. log(A^x)=x log(A)

    Applying law 1 we have:

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437)
    ...........= log(187^273) + log(456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437)
    ...........= log(187^273) + log(456^135) + log( 299^424 x 588^139 x 394^154 x 181^177 x 1342^437)

    and so on to get:

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437)
    ...........= log(187^273) + log(456^135) + log(299^424) + log(588^139) + log(394^154) + log(181^177) + log(1342^437).

    Now apply law 2 to each term on the righthand side of the "=" sign
    to get the result given earlier:

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437) =
    .........273 log(187) + 135 log(456) + 424 log(299) + 139 log(588) + 154 log(394) + 177 log(181) + 437 log(1342).

    Now this can be evaluated on a calculator, to (if I have done this right) 4579.9444.

    But by the definition of common logarithms if log(a)=b, then
    10^b=a. Now we have:

    log(187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437) = 4579.9444,

    so:

    187^273 x 456^135 x 299^424 x 588^139 x 394^154 x 181^177 x 1342^437
    .........~= 10^4579.9444 = 10^4579 x 10^0.9444
    .........~= 8.799 x 10^4579.

    RonL
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