Simplifying fractional exponents with variables

Question

The problem

I do not understand how the exponents are simplified to

$\frac pq-1$

in the expression in the image

Answer 1

Well in the interest of closing this out:

The basic rules of exponents state that

$x^ax^b = x^{a+b}$ and
$\frac{x^a}{x^b} =x^{a-b}$

We start by requiring that $a$ and $b$ be integers but later on realize this works for rational numbers such as $p/q$ in your example and even all real numbers.

In your example, we can ignore $p/q$ on the left in each expression and apply the second rule above:

$\frac{x^p}{x^{p-p/q}} = x^{p-1-(p-p/q)} = x^{p/q-1}$

As an aside, the RHS of the original sure looks like a derivative but I can't see where the LHS came from.