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Math Help - tricky negative simplification

  1. #1
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    tricky negative simplification

    This is a interesting problem:

    (-5)^2 * -5^3

    = -5^2+3

    = -5^5

    Using the calculator, both (-5)^2 * -5^3 and -5^5 is -3125.

    Now, consider this:

    (-2)^3 * 2^5

    = -2^3+5

    = -2^8

    Using the calculator again, we have different results:

    (-2)^3 * 2^5 = -8 * 32 = -256

    -2^8 = 256

    The original is negative and the simplified expression is positive. It looks like the expression cannot be simplified, but why??

    Please help. Thanks.
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  2. #2
    MHF Contributor Quick's Avatar
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    the rule: a^x*a^y=a^(x+y)

    is only guaranteed to work when a equals a

    notice: (-a)^x*a^y

    -a doesn't equal a

    therefore you can't add the exponents.
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  3. #3
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    Quote Originally Posted by Quick View Post

    -a doesn't equal a

    Thanks again, Math can be a lot easier without the negatives.
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by shenton View Post
    This is a interesting problem:

    (-5)^2 * -5^3

    = -5^2+3

    = -5^5

    Using the calculator, both (-5)^2 * -5^3 and -5^5 is -3125.

    Now, consider this:

    (-2)^3 * 2^5

    = -2^3+5

    = -2^8

    Using the calculator again, we have different results:

    (-2)^3 * 2^5 = -8 * 32 = -256

    -2^8 = 256

    The original is negative and the simplified expression is positive. It looks like the expression cannot be simplified, but why??

    Please help. Thanks.
    There are two things going on here. First, Quick is correct in his post, if the bases are not the same you can't add the exponents. Also, as is mentioned in another recent post when you put -2^8 in your calculator you will automatically get a positive result. Of course, with the way you intended the expression should get one in this case, but the expression -2^8 as written is -(2^8) not (-2)^8. So be careful and use parenthesis!

    To get around the problem mentioned by Quick, what I would recommend is this:

    (-2)^3 = (-1)^3 * 2^3 = - 2^3.

    So (-2)^3 * 2^5 = (-1)^3 * 2^3 * 2^5 = -1 * 2^8 = -256.

    This both allows you to simplify the problem by adding exponents, and also highlights the pesky "-" out in front.

    -Dan
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  5. #5
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    Quote Originally Posted by topsquark View Post

    To get around the problem mentioned by Quick, what I would recommend is this:

    (-2)^3 = (-1)^3 * 2^3 = - 2^3.

    So (-2)^3 * 2^5 = (-1)^3 * 2^3 * 2^5 = -1 * 2^8 = -256.

    This both allows you to simplify the problem by adding exponents, and also highlights the pesky "-" out in front.
    This is awesome! One must be a genious to think of the problem in the above manner. Thanks!
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by shenton View Post
    This is awesome! One must be a genious to think of the problem in the above manner. Thanks!
    Thanks for recognizing my ability.

    Seriously, I'm just a guy that has made the same mistakes and had someone else show me this stuff.

    (Whispers) Pass it on!

    -Dan
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