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Let's simplify the given expression \(\left(\frac{-6d y^6}{-6d^6 y^8}\right)^2\).

### Step-by-Step Simplification:
1. **Simplify the fraction inside the parentheses:**
   \[
   \frac{-6d y^6}{-6d^6 y^8}
   \]
   - The \(-6\) terms in the numerator and denominator cancel each other out.
   - The \(d\) in the numerator cancels with one \(d\) in the denominator, leaving \(d^{5}\) in the denominator.
   - For the \(y\) terms, \(y^6\) in the numerator and \(y^8\) in the denominator reduce to \(\frac{y^6}{y^8} = y^{-2}\).

   After simplification, we get:
   \[
   \frac{1 \cdot y^{-2}}{d^{5}} = \frac{y^{-2}}{d^{5}}
   \]

2. **Rewrite the simplified fraction:**
   \[
   \frac{y^{-2}}{d^{5}} = \frac{1}{d^{5} y^{2}}
   \]

3. **Square the result:**
   \[
   \left( \frac{1}{d^{5} y^{2}} \right)^2 = \frac{1}{(d^{5} y^{2})^2} = \frac{1}{d^{10} y^{4}}
   \]

So, the final simplified form of the given expression is:

\[
\frac{1}{d^{10} y^{4}}
\]

Would you like more details on any of these steps? Here are some related questions for further exploration:

1. What is the general rule for dividing variables with exponents?
2. How do you handle negative exponents in simplification?
3. What happens when you multiply or divide variables with coefficients?
4. How do you simplify expressions with multiple variables?
5. What are some common mistakes to watch out for in exponent rules?

**Tip:** Always simplify the terms inside parentheses before applying any exponents outside the parentheses to avoid unnecessary complexity.

Simplify the expression \(\left(\frac{-6d y^6}{-6d^6 y^8}\right)^2\).

Learn how to simplify the expression \(\left(\frac{-6d y^6}{-6d^6 y^8}\right)^2\) using the quotient rule and exponent laws. A step-by-step solution is provided, suitable for high school students.

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