How to expand and simplify $(p^y * (pq)^o) / (p^{(2y+o)} * q^{(o-2)})$?

Question

I'm a beginner to maths and have trouble simplyfying the following function:

$$\frac{p^y \cdot (pq)^o}{p^{2y+o} \cdot q^{o-2}}$$

The final answer is

$$p^{-y} \cdot q^2$$

But I'm not sure how to get there.

Any help is appreciated.

Answer 1

Here's the method in general, without actually working out your example. You should do that yourself to seal the concepts.

The intermediate goal is to get all the powers of p and q separated in both the numerator and denominator. In this case, it's almost there, with the exception of (pq)^o. So expand that first, using the principle (x*y)^a = x^a * y^a.

Then gather the p's and q's using the properties of multiplication and exponentiation, x^a * x^b = x^(a+b). Finally, match the p's in the numerator and denominator, likewise the q's, and using the principle x^a / x^b = x^(a-b) calculate the ultimate powers of p and q. Note that there are two ways you could handle the power of p in the final answer, since it is negative.