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The given expression is:

\[
\frac{3y^{-1} + 4x^{-1}}{9y^{-2} - 16x^{-2}}
\]

### Step 1: Simplify the exponents

- \( y^{-1} = \frac{1}{y} \), and \( x^{-1} = \frac{1}{x} \).
- Similarly, \( y^{-2} = \frac{1}{y^2} \) and \( x^{-2} = \frac{1}{x^2} \).

So the expression becomes:

\[
\frac{\frac{3}{y} + \frac{4}{x}}{\frac{9}{y^2} - \frac{16}{x^2}}
\]

### Step 2: Combine terms in the numerator

For the numerator:

\[
\frac{3}{y} + \frac{4}{x} = \frac{3x + 4y}{xy}
\]

Thus, the numerator is:

\[
\frac{3x + 4y}{xy}
\]

### Step 3: Combine terms in the denominator

The denominator is:

\[
\frac{9}{y^2} - \frac{16}{x^2}
\]

This is a difference of squares, which can be factored as:

\[
\frac{9}{y^2} - \frac{16}{x^2} = \frac{(3/y - 4/x)(3/y + 4/x)}
\]

Thus, the denominator becomes:

\[
\frac{(3x - 4y)(3x + 4y)}{x^2 y^2}
\]

### Step 4: Combine the full expression

Now, divide the numerator by the denominator:

\[
\frac{\frac{3x + 4y}{xy}}{\frac{(3x - 4y)(3x + 4y)}{x^2 y^2}}
\]

This simplifies by multiplying by the reciprocal of the denominator:

\[
\frac{3x + 4y}{xy} \times \frac{x^2 y^2}{(3x - 4y)(3x + 4y)} = \frac{xy \cdot x^2 y^2}{(3x - 4y)(3x + 4y) \cdot xy}
\]

Cancel out common factors:

\[
\frac{1}{3x - 4y}
\]

### Final Answer:

The simplified expression is:

\[
\boxed{\frac{1}{3x - 4y}}
\]

Would you like further clarification or details on any step?

### 5 Related Questions:
1. How do you handle negative exponents in general?
2. Can you apply this method to other rational expressions involving exponents?
3. What happens if \( x = y \) in the final simplified expression?
4. How does factoring help in simplifying rational expressions?
5. Can this expression be simplified further if specific values for \( x \) and \( y \) are known?

### Tip:
When simplifying fractions with exponents, always try to convert negative exponents into fractions to make the algebra easier.

Simplify the expression (3y^-1 + 4x^-1) / (9y^-2 - 16x^-2).

Learn how to simplify the expression (3y^-1 + 4x^-1) / (9y^-2 - 16x^-2) by applying exponent rules and factoring. This step-by-step solution is perfect for students in Grades 10-12, covering algebraic techniques like combining rational expressions and difference of squares.

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