How to prove (a^c)^d = a^(cd) without knowing the values of c and d?

[Llamas]

Homework Statement

The problem is in the title: Prove (a^c)^d) = a^(cd)

Homework Equations

N is the set of natural numbers.

(ab)^n = (a^n)(b^n)
a^(p+n) = (a^p)*(a^n)
((a^p)(a^n)) * a = (a^p)(a^(n+1))

The Attempt at a Solution

c = p-q; d = j-k; p,q,j,kεN (by definition of integers)

(a^(p-q))^(j-k)

((a^(p-q))^j)/((a^(p-q))^k)

((a^(p+(-q)))^j)/((a^(p+ (-q)))^k)

((a^p)(a^(-q)))^j/((a^p)(a^(-q)))^k or (((a^p)/(a^q))^j)/(((a^p)/(a^q))^k)

I have no idea where to go from here. I've spent two hours on this and I've finished nearly all my other proofs on this assignment...This one is killing me though, and I have other work I need to get to at some point. Could anyone please give me a tip in the right direction? I feel like I have it all wrong.

 

[tiny-tim]

Welcome to PF!

Hi Llamas! Welcome to PF!

What are c and d, are they all natural (whole) numbers?

if so, just write a^c = a times a^c-1

 

[Llamas]

Hi Llamas! Welcome to PF!

What are c and d, are they all natural (whole) numbers?

if so, just write a^c = a times a^c-1

Thank you tiny-tim, the advice helped me finish the proof. Sorry I didn't respond until now, had other work as well.