Tough algebra problem. Fractions and negative exponents

This problem with fractions and negative exponents is killing me.. I thought algebra was supposed to be easy, lol. ;) Any help greatly appreciated. I know the answer but cant get the steps right for some reason..

Simplify: Assume that all variables represent nonzero integers.

$\displaystyle

(2^-2)^a x (2^b)^-a / (2^-2)^-b x (2^b)^-2a

$

Here is the answer according to the book:

Ans. = 2^(-2a - 2b + ab)

I'm never able to get the final answer correct for some reason. I know it's hard to read these problems using computer text but I'd appreciate if you can bare with me and tell me where I'm messing up here.

Original Numerator: (2^-2)^a x (2^b)^-a

= 2^-2a x 2^-ba

= 1/(4^a) x 1/(2^ba)

= (1 / 8^2ab) **<--simplified numerator **

Original Denominator: (2^-2)^-b x (2^b)^-2a

= 2^2b x 2^-2ab

= 2^2b x 1/(2^ab)

= (2^2b / 2^ab) **<--simplified denominator**

So the new problem is:

(1 / 8^2ab) / (2^2b / 2^ab)

Every time I solve this with cross multiplication I get the wrong answer...so I'm guessing I messed up somewhere along the way.. Please heeelp!

;)

I've tried everything I know but never seem to get the correct answer. I know typing formulas is a pain, but if someone could show me the steps involved I'd really appreciate it!