(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

x is in the arbitrary multiplicative group, and a,b are positive integers.

given that

[tex] x^{a+b} = x^ax^b[/tex] and [tex] (x^a)^b =x ^{ab} [/tex]

show

that

[tex] (x^a)^-1 = x^{-a} [/tex]

2. Relevant equations

na

3. The attempt at a solution

Induction:

I) (x)^{-1} = x ^{-1}

II) Assume [tex] (x^n)^{-1} = x^{-n} [/tex], to prove that [tex] (x^{n+1})^{-1} = x^{-(n+1)} [/tex].

[tex] (x^{n+1})^{-1}) = (x^nx)^{-1} = x^{-1}x^{-n} = x^{-n-1} [/tex]

Is the last step justified?

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# Homework Help: Group inverses

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