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Thread: Confusing indices rules and the meanings behind them.

  1. #1 Confusing indices rules and the meanings behind them. 
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    Hello guys, i just wanted to ask the mathematical proof that x to the negative power of 1= one over x. please answer with full mathematical proof behind it.

    merci beaucoup!


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  3. #2  
    Forum Junior anticorncob28's Avatar
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    There are a few ways that you can see this.
    First, note that (a^b)/(a^c) = a^(b - c). This works out because if you multiply it out, you get (a*a*a... b times)/(a * a * a... c times), and all the c terms cancel on the bottom, and there are (b - c) terms on the top. Now, we can compute 1/a = a/(a^2) = (a^1)/(a^2) = a^-1.
    You can also look at this like a pattern (though this is similar to the above proof, and this is more inductive reasoning): Let's list out the powers of two:
    2^0 2^1 2^2 2^3 ...
    1 2 4 8 ...
    When you keep going to the right, you keep multiplying by two, and when you go to the left, you keep dividing by two. If you continue the pattern to the left, you get 2^-1 = 1/2.


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  4. #3  
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    There is no "proof" of it, it is a definition. But there are reasons for the definition being that way. This is what anticorncob28 is talking about.

    Start by defining , for a any positive number and n a positive integer to be "a multiplied by itself n times" (, , etc.). It is easy to see from that the "property of exponents", , by "counting" the number of "a"s on each side.

    That's a very nice property and we would like it to be true as we extend to other exponents. We would like to be able to say . Since n+ 0= n, that is the same as which is just saying "". Similarly, every negative integer can be written as "-n" for some positive integer n: that is, n+ (-n)= 0. If we want then we must have so we must define and, in particular, if n= 1, .
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  5. #4  
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    Well, it is kind of a definition. In general, you want a quantity which is the inverse for x which means that



    where e is the neutral element of your operation under consideration. For addition, your e = 0 and your inverse is written -x. For multiplication you have 1*x = x, so 1 is your neutral element, e=1. The rest follows from laws of multiplication and division and the fact that the inverse is unique.
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