Let's simplify the expression:

\[
\frac{\frac{x}{y^2} + \frac{1}{y}}{\frac{y}{x^2} + \frac{1}{x}}
\]

### Step 1: Simplify the numerator

The numerator is:

\[
\frac{x}{y^2} + \frac{1}{y}
\]

To combine these into a single fraction, find a common denominator:

\[
\frac{x}{y^2} + \frac{1}{y} = \frac{x + y}{y^2}
\]

### Step 2: Simplify the denominator

The denominator is:

\[
\frac{y}{x^2} + \frac{1}{x}
\]

Similarly, find a common denominator:

\[
\frac{y}{x^2} + \frac{1}{x} = \frac{y + x}{x^2}
\]

### Step 3: Rewrite the entire expression

Now, the original expression becomes:

\[
\frac{\frac{x + y}{y^2}}{\frac{y + x}{x^2}}
\]

### Step 4: Simplify the expression

This is a division of two fractions, so we multiply by the reciprocal:

\[
\frac{x + y}{y^2} \times \frac{x^2}{y + x}
\]

Since \( x + y \) and \( y + x \) are the same, they cancel out, and we are left with:

\[
\frac{x^2}{y^2}
\]

Thus, the simplified expression is:

\[
\frac{x^2}{y^2} = \left(\frac{x}{y}\right)^2
\]

Would you like more details on any of these steps or have any questions?

### Relative Questions:
1. How do you simplify complex rational expressions?
2. What is the process for finding a common denominator?
3. How do you factor and simplify algebraic expressions?
4. What are some common mistakes to avoid when simplifying expressions?
5. How does this problem relate to the concept of ratios and proportions?

### Tip:
When simplifying complex fractions, always look for opportunities to cancel out common terms after simplifying the numerator and denominator separately.

Learn how to simplify the complex rational expression (x/y^2 + 1/y) / (y/x^2 + 1/x) with detailed step-by-step instructions. This problem covers rational expressions, fraction simplification, and finding common denominators. Suitable for Grades 9-12 students.

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