We are given the expression:

$\frac{\frac{2x}{y^{2}}+\frac{3y}{x^{3}}}{4-\frac{7}{xy}}$

Let’s break it down step by step.

Step 1: Work on the numerator

The numerator is: $\frac{2x}{y^2} + \frac{3y}{x^3}$

There are no common factors between these two terms, so we leave it as it is for now.

Step 2: Work on the denominator

The denominator is: $4 - \frac{7}{xy}$

Again, there is no common factor between these terms, so the expression remains as is.

Step 3: Combine the expression

So, the full expression is: $\frac{\frac{2x}{y^2} + \frac{3y}{x^3}}{4 - \frac{7}{xy}}$

This is the simplified form of the given expression.

Do you want me to perform additional operations, or simplify further?

Related Questions:

1. How can we factor expressions like $\frac{2x}{y^2} + \frac{3y}{x^3}$?
2. What is the common denominator of $\frac{2x}{y^2}$ and $\frac{3y}{x^3}$?
3. How would the expression change if we introduced a substitution such as $u = \frac{x}{y}$?
4. Can we rewrite the denominator $4 - \frac{7}{xy}$ using partial fractions?
5. What happens if $xy = 1$ in the denominator?

Tip:

When simplifying fractions with variables, always check for common factors or opportunities to factor expressions to make the process easier.