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**Simplify the following expressions using the law of exponents**

The given expression is:

\[
(2x^2)^{-4}
\]

### Simplification Using the Laws of Exponents:

1. **Apply the exponent to both the constant and the variable:**
   Using the law of exponents \((ab)^n = a^n \cdot b^n\), we distribute the exponent \(-4\) to both \(2\) and \(x^2\).

   \[
   (2x^2)^{-4} = 2^{-4} \cdot (x^2)^{-4}
   \]

2. **Simplify each term:**
   - \(2^{-4} = \frac{1}{2^4} = \frac{1}{16}\)
   - \((x^2)^{-4} = x^{2 \cdot -4} = x^{-8}\)

3. **Combine the terms:**
   \[
   \frac{1}{16} \cdot x^{-8} = \frac{x^{-8}}{16}
   \]

4. **Rewrite with positive exponents (optional):**
   Since \(x^{-8}\) can be rewritten as \(\frac{1}{x^8}\), the expression becomes:
   \[
   \frac{1}{16x^8}
   \]

### Final Simplified Expression:

\[
\frac{1}{16x^8}
\]

Would you like to explore more about this process or ask any additional questions?

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Here are some related questions for further exploration:
1. How do you apply exponents to both constants and variables within parentheses?
2. What is the effect of raising a power to a negative exponent?
3. How do you simplify expressions with multiple exponents applied to one term?
4. What are the rules for handling fractional exponents within a simplification problem?
5. How does simplifying negative exponents differ from positive exponents?

**Tip:** When distributing negative exponents across parentheses, ensure that you handle both constants and variables individually before combining them.

Learn how to simplify the expression (2x^2)^{-4} using the laws of exponents with a detailed step-by-step solution. This problem covers key mathematical concepts in exponents and is suitable for students in Grades 7-9.

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