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

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.

Learn how to simplify the complex rational expression \frac{\frac{2x}{y^2} + \frac{3y}{x^3}}{4 - \frac{7}{xy}} with step-by-step guidance. This solution involves algebraic concepts such as operations on rational expressions and simplifying fractions. Suitable for students in Grades 9-12.

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