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Math Help - How can you rewrite powers like this: a^x - b^x?

  1. #1
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    How can you rewrite powers like this: a^x - b^x?

    Hi there,
    I am proving something, and I want to know how to rewrite a formula like this one: a^x - b^x, where you know that b<a.
    Is there any way to write this differently? Maybe with sumformula, or an easy powerrule...anything?
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  2. #2
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    Quote Originally Posted by MaryB View Post
    Hi there,
    I am proving something, and I want to know how to rewrite a formula like this one: a^x - b^x, where you know that b<a.
    Is there any way to write this differently? Maybe with sumformula, or an easy powerrule...anything?
    There is no "rule of exponents" that will simplify that.

    IF x is a positive integer, then a^x- b^x= (a- b)(a^{x-1}+ a^{x-2}b+ a^{x-3}b^2+ \cdot\cdot\cdot+ a^2b^{x-3}+ ab^{x-2}+ b^{x-1}) but I doubt that is what you want.
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  3. #3
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    Actually, that is a very useful rewriting you gave me there.
    Does it have a name? Or is it deduced from another formula?
    I think I can use it!
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  4. #4
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    Hello,

    See here : http://www.mathhelpforum.com/math-he...484-k-b-k.html for a way to prove it
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  5. #5
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    Quote Originally Posted by Moo View Post
    Hello,
    See here : http://www.mathhelpforum.com/math-he...484-k-b-k.html for a way to prove it
    Brilliant thanks! Btw, I love your avatar.
    Okay well, my assignment is getting near the solution, but not yet. So I now have (a- b)(a^{x-1}+ a^{x-2}b+ a^{x-3}b^2+ \cdot\cdot\cdot+ a^2b^{x-3}+ ab^{x-2}+ b^{x-1}) and found out that if a = a+1, then b = a. so...now I have (a+1 -a)((a+1)^{x-1}+ (a+1)^{x-2}a+ (a+1)^{x-3}a^2+ \cdot\cdot\cdot+ (a+1)^2a^{x-3}+ (a+1)a^{x-2}+ a^{x-1})
    And yeah, okay. How can I rewrite this one? I don't think it's neccesary to give the whole rewriting, but a hint? (Or the whole rewriting is okay, I can stop myself from looking first and try it out myself).
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  6. #6
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    Quote Originally Posted by MaryB View Post
    (a+1 -a)((a+1)^{x-1}+ (a+1)^{x-2}a+ (a+1)^{x-3}a^2+ \cdot\cdot\cdot+ (a+1)^2a^{x-3}+ (a+1)a^{x-2}+ a^{x-1})
    And yeah, okay. How can I rewrite this one? I don't think it's neccesary to give the whole rewriting, but a hint? (Or the whole rewriting is okay, I can stop myself from looking first and try it out myself).
    Does anyone know if I can rewrite in a way it shows a sumformula? Because I think that is quite possible, but I don't really know..how!
    Please help
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