https://file.aiquickdraw.com/mathbot/file/17268860443899zo5ni45.jpg

The problem in the image asks to **simplify** the following expression:

\[
\frac{\left(\frac{x^5}{6x}\right)^{-2}}{\left(\frac{x^4 y^2}{(y-6)^3}\right)}
\]

### Step-by-step simplification:

1. **Simplify the numerator inside the first fraction:**
   \[
   \frac{x^5}{6x} = \frac{x^5}{x^1} = x^{5-1} = x^4
   \]
   So the expression becomes:
   \[
   \frac{\left(x^4\right)^{-2}}{\left(\frac{x^4 y^2}{(y-6)^3}\right)}
   \]

2. **Apply the exponent on \(x^4\):**
   \[
   (x^4)^{-2} = x^{4 \cdot (-2)} = x^{-8}
   \]
   So now the expression becomes:
   \[
   \frac{x^{-8}}{\frac{x^4 y^2}{(y-6)^3}}
   \]

3. **Rewrite the complex fraction:**
   \[
   \frac{x^{-8}}{\frac{x^4 y^2}{(y-6)^3}} = x^{-8} \cdot \frac{(y-6)^3}{x^4 y^2}
   \]

4. **Simplify the powers of \(x\):**
   \[
   x^{-8} \cdot \frac{(y-6)^3}{x^4 y^2} = \frac{(y-6)^3}{x^{8+4} y^2} = \frac{(y-6)^3}{x^{12} y^2}
   \]

So the simplified expression is:

\[
\boxed{\frac{(y-6)^3}{x^{12} y^2}}
\]

Would you like further clarification on any of the steps or more details?

Here are five related questions to expand your understanding:
1. How do you handle exponents in fractions when simplifying?
2. What are the rules for dealing with negative exponents?
3. How does multiplying powers of the same base work?
4. Can you apply these techniques to simplify radical expressions?
5. How would the expression change if we added terms inside the parentheses?

**Tip:** Always simplify fractions and exponents step by step to avoid mistakes, especially when dealing with complex expressions.

Simplify the following expression: ((x^5) / (6x))^(-2) / (x^4 * y^2 / (y-6)^3)

Learn how to simplify a complex expression with negative exponents and fractions. This detailed guide covers exponent rules, fraction simplification, and algebraic manipulation. Suitable for high school students in Grades 10-12.

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