To solve the expression \((7m^{-3}n^{8})^{-2}\), we'll apply the power rule for exponents, which states that \((a^m)^n = a^{mn}\).

### Step 1: Apply the Power Rule to Each Factor
\[
(7m^{-3}n^{8})^{-2} = 7^{-2} \cdot (m^{-3})^{-2} \cdot (n^{8})^{-2}
\]

### Step 2: Simplify Each Factor
- For \(7^{-2}\), this is simply the reciprocal:
  \[
  7^{-2} = \frac{1}{7^2} = \frac{1}{49}
  \]
- For \((m^{-3})^{-2}\), multiply the exponents:
  \[
  (m^{-3})^{-2} = m^{(-3) \cdot (-2)} = m^{6}
  \]
- For \((n^{8})^{-2}\), multiply the exponents:
  \[
  (n^{8})^{-2} = n^{8 \cdot (-2)} = n^{-16}
  \]

### Step 3: Combine the Results
\[
\frac{1}{49} \cdot m^{6} \cdot n^{-16}
\]
This can be written as:
\[
\frac{m^6}{49n^{16}}
\]

So, the 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 exponents were different?
2. What happens when the base is negative?
3. How do you simplify an expression with fractional exponents?
4. How can you apply this to solve real-world problems involving exponents?
5. How does this apply to polynomial expressions?

**Tip:** When dealing with negative exponents, remember that \(a^{-n} = \frac{1}{a^n}\).

Learn how to simplify the expression (7m^-3n^8)^-2 step-by-step using the power rule for exponents. This problem covers exponents, negative exponents, and fractional exponents, suitable for high school students.

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