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Let's solve the problems step by step:

### (a) Simplify \((2x^{-6})^2\):

1. **Apply the exponent to both terms inside the parentheses**:  
   \[
   (2x^{-6})^2 = 2^2 \cdot (x^{-6})^2
   \]
   
2. **Simplify the exponents**:  
   \[
   2^2 = 4
   \]  
   For \( (x^{-6})^2 \), use the power rule \( (a^m)^n = a^{m \cdot n} \):  
   \[
   (x^{-6})^2 = x^{-12}
   \]
   
   So the expression becomes:  
   \[
   4x^{-12}
   \]

3. **Eliminate the negative exponent by writing it as a reciprocal**:  
   \[
   4x^{-12} = \frac{4}{x^{12}}
   \]

   **Final answer** for (a):  
   \[
   \frac{4}{x^{12}}
   \]

---

### (b) Simplify \(\frac{x^5y^5}{x^8y^{-8}}\):

1. **Use the quotient rule for exponents**:  
   \[
   \frac{x^5}{x^8} = x^{5-8} = x^{-3}
   \]  
   \[
   \frac{y^5}{y^{-8}} = y^{5 - (-8)} = y^{5 + 8} = y^{13}
   \]

2. **Simplify**:  
   \[
   x^{-3}y^{13}
   \]

3. **Eliminate the negative exponent**:  
   \[
   x^{-3}y^{13} = \frac{y^{13}}{x^3}
   \]

   **Final answer** for (b):  
   \[
   \frac{y^{13}}{x^3}
   \]

---

Would you like further clarification on any part of the solution?

Here are some related questions to deepen your understanding:

1. How would you simplify \((3x^{-2})^3\)?
2. What is the rule for multiplying two terms with the same base but different exponents?
3. How do you simplify \(\frac{a^{-4}}{a^{-7}}\)?
4. If given \((xy^3)^4\), how would you simplify it?
5. What is the result of \(\frac{x^2y^{-3}}{x^{-1}y^5}\)?

**Tip**: Always apply exponent rules step by step, and remember to simplify negative exponents by rewriting them as fractions.

Simplify the following and express the answer using no negative exponents.
(a) (2x^{-6})^2
(b) \frac{x^5y^5}{x^8y^{-8}}

Learn how to simplify expressions involving exponents, such as (2x^{-6})^2 and \frac{x^5y^5}{x^8y^{-8}}, using the laws of exponents. This step-by-step solution covers exponent rules and how to handle negative exponents, suitable for Grades 8-10.

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