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To solve the problem, we need to simplify the expression:

\[
\frac{\frac{36a}{b} - \frac{b}{a}}{\frac{1}{a} - \frac{6}{b}}
\]

Let's follow the steps:

1. **Simplify the numerator**:
   \[
   \frac{36a}{b} - \frac{b}{a}
   \]
   Find a common denominator, which is \(ab\):
   \[
   \frac{36a^2}{ab} - \frac{b^2}{ab} = \frac{36a^2 - b^2}{ab}
   \]

2. **Simplify the denominator**:
   \[
   \frac{1}{a} - \frac{6}{b}
   \]
   Find a common denominator, which is \(ab\):
   \[
   \frac{b}{ab} - \frac{6a}{ab} = \frac{b - 6a}{ab}
   \]

3. **Combine the simplified numerator and denominator**:
   \[
   \frac{\frac{36a^2 - b^2}{ab}}{\frac{b - 6a}{ab}}
   \]
   When dividing by a fraction, multiply by its reciprocal:
   \[
   \frac{36a^2 - b^2}{ab} \times \frac{ab}{b - 6a} = \frac{36a^2 - b^2}{b - 6a}
   \]

4. **Factorize the numerator** (if possible):
   \[
   36a^2 - b^2 = (6a)^2 - b^2 = (6a - b)(6a + b)
   \]

5. **Simplify the final expression**:
   \[
   \frac{(6a - b)(6a + b)}{b - 6a}
   \]
   Notice that \(b - 6a = -(6a - b)\):
   \[
   \frac{(6a - b)(6a + b)}{-(6a - b)} = -(6a + b)
   \]

Therefore, the simplified expression is:
\[
-(6a + b)
\]

Do you want any details or have any questions about this solution? 

Here are 5 related questions to expand your understanding:

1. How do you simplify complex rational expressions with multiple fractions?
2. What are some common techniques for factoring polynomial expressions?
3. How do you find the least common denominator for algebraic fractions?
4. Can you explain the importance of recognizing patterns in algebraic expressions?
5. What is the difference between rationalizing the numerator and the denominator in fractions?

**Tip:** Always look for common factors and patterns to simplify expressions more efficiently.

Simplify \( \frac{\frac{36a}{b} - \frac{b}{a}}{\frac{1}{a} - \frac{6}{b}} \).

Learn how to simplify the rational expression \( \frac{\frac{36a}{b} - \frac{b}{a}}{\frac{1}{a} - \frac{6}{b}} \) by following step-by-step simplification techniques, including finding common denominators and factoring. Suitable for students in Grades 10-12.

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