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 $(pq)^0$. So expand that first, using the principle $(x\times y)^a = x^a \times y^a$.

Then gather the $p$'s and $q$'s using the properties of multiplication and exponentiation, $x^a \times x^b = x^{a+b}$. Finally, match the $p$'s in the numerator and denominator, likewise the $q$'s, and using the principle $\frac{x^a}{x^b} = x^{a-b}$ 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.