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.