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## Homework Statement

Prove:

[itex]

$\left(ab^{-1}\right)\left(cd^{-1}\right)^{-1}=\left(ad\right)\left(cb\right)^{-1}$.

[/itex]

I know how to distribute exponents to get both sides to look identical...

## Homework Equations

...but a step in the solution requires distributing exponents. But how do you prove that you can distribute exponents using only:

- association for addition and multiplication
- commutation for addition and multiplication
- identity for addition and multiplication
- additive and multiplicative inverses
- distribution
- trichotomy law
- closure for addition and multiplication

## The Attempt at a Solution

This problem is presented before mathematical induction is introduced, so I would like to solve it without its use. But here is my attempt at induction:

P(m) = (a^n)^m=a^(n*m)

P(1) = (a^n)^1=a^n

P(k+1)=(a^n)^(k+1)=a^(n*k+n*1)