To simplify \[ (-8)^{-\frac{2}{3}} \]:

### Step-by-Step Solution:
1. **Apply the Negative Exponent Rule:**  
   A negative exponent, \( a^{-n} = \frac{1}{a^n} \).
   \[
   (-8)^{-\frac{2}{3}} = \frac{1}{(-8)^{\frac{2}{3}}}
   \]

2. **Rewrite with a Fractional Exponent:**  
   The exponent \( \frac{2}{3} \) means to take the cube root first and then square.
   \[
   \frac{1}{(-8)^{\frac{2}{3}}} = \frac{1}{\left((-8)^{\frac{1}{3}}\right)^2}
   \]

3. **Find the Cube Root of -8:**  
   The cube root of \(-8\) is \(-2\) because \((-2)^3 = -8\).
   \[
   \frac{1}{\left((-8)^{\frac{1}{3}}\right)^2} = \frac{1}{(-2)^2}
   \]

4. **Square the Result:**  
   \((-2)^2 = 4\), so we have:
   \[
   \frac{1}{4}
   \]

### Final Answer:
\[
(-8)^{-\frac{2}{3}} = \frac{1}{4}
\]

Would you like further explanation on any step?

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Here are 5 related questions to deepen understanding:

1. What is the result of simplifying \( 27^{-\frac{2}{3}} \)?
2. How do you simplify expressions with exponents like \( \frac{1}{a^{\frac{3}{4}}} \)?
3. What is the cube root of \( -64 \)?
4. How do you handle fractional exponents with different bases, such as \( (-27)^{\frac{1}{3}} \)?
5. How can you simplify \((-125)^{\frac{2}{3}}\) using similar steps?

**Tip:** For any base raised to a fractional exponent, remember that the numerator is the power, and the denominator is the root.

Discover how to simplify the expression (-8)^(-2/3) by applying rules for negative and fractional exponents. Follow a step-by-step solution that includes taking the cube root and squaring the result, suitable for Grades 9-11 students learning about exponents.

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