Let's simplify the expression \((7m^{-3}n^{8})^{-2}\) step by step.

### Step 1: Apply the Power Rule to Each Factor
The expression is \((7m^{-3}n^{8})^{-2}\). Using the power rule \((a^m)^n = a^{mn}\), distribute the exponent \(-2\) to each factor inside the parentheses:

\[
7^{-2} \cdot (m^{-3})^{-2} \cdot (n^{8})^{-2}
\]

### Step 2: Simplify Each Factor
- For \(7^{-2}\):
  \[
  7^{-2} = \frac{1}{7^2} = \frac{1}{49}
  \]
- For \((m^{-3})^{-2}\):
  \[
  (m^{-3})^{-2} = m^{6}
  \]
- For \((n^{8})^{-2}\):
  \[
  (n^{8})^{-2} = n^{-16}
  \]

### Step 3: Combine the Results
Now, multiply these simplified results together:

\[
\frac{1}{49} \cdot m^{6} \cdot n^{-16}
\]

This simplifies to:

\[
\frac{m^6}{49n^{16}}
\]

So, the fully simplified expression is:

\[
\frac{m^6}{49n^{16}}
\]

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would the expression change if the exponent was \(-3\) instead of \(-2\)?
2. What happens if the base includes a negative number?
3. How do you simplify expressions with fractional exponents?
4. What is the significance of a negative exponent in real-world contexts?
5. How do you simplify an expression with multiple variables and exponents?

**Tip:** When simplifying expressions with exponents, always apply the power rule carefully to avoid mistakes.

Learn how to simplify the expression (7m^-3n^8)^-2 step-by-step with detailed explanations. This problem involves applying the power rule to exponents and simplifying complex expressions. Suitable for advanced high school students.

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