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Math Help - Equalities

  1. #1
    MHF Contributor harish21's Avatar
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    Equalities

    For a non-zero x and y, and integers a and b, prove the following inequalities:

    (1) x^(a+b) = x^a * x^b

    (2) (xy)^a = x^a * y^a

    can anyone give me a hint on how to prove these equalities
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  2. #2
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    Try using logarithms. Here is a quick refresher: http://en.wikipedia.org/wiki/Logarithm.

    I hope that's sufficient.
    Last edited by simulacrum; February 21st 2010 at 06:39 PM. Reason: Fixed link.
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by simulacrum View Post
    Try using logarithms. Here is a quick refresher: Logarithm - Wikipedia, the free encyclopedia.

    I hope that's sufficient.
    Thanks. I wanted to know if I had to use induction here..
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  4. #4
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    Quote Originally Posted by harish21 View Post
    For a non-zero x and y, and integers a and b, prove the following inequalities:

    (1) x^(a+b) = x^a * x^b

    (2) (xy)^a = x^a * y^a

    can anyone give me a hint on how to prove these equalities
    Using logarithms seems to me to be overkill here. If a and b are positive integers, then x^(a+b) means x multiplied by itself a+ b times. But, using the associative law to move parentheses, that is the same as x multiplied by itself a times and x multiplied by itself b times: x^ax^b.

    Similarly, (xy)^a means xy multiplied by itself a times. Use the commutative law to write that as x multiplied by itself a times and y multiplied by itself a times: x^ay^a.

    For a or b positive, use the fact that x^0= 1.

    For a or b negative, use the fact that x^(-a)= 1/x^(a).
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