The expression \( 8^{-\frac{2}{3}} \) can be simplified as follows:

1. **Rewrite the expression**: The exponent can be broken down into two parts:
   \[
   8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}}
   \]

2. **Evaluate the cube root**: \( 8^{\frac{1}{3}} \) represents the cube root of 8. Since \( 8 = 2^3 \), the cube root of 8 is 2:
   \[
   8^{\frac{1}{3}} = 2
   \]

3. **Square the result**: Now, square the result from the previous step:
   \[
   8^{\frac{2}{3}} = 2^2 = 4
   \]

4. **Take the reciprocal**: Since the original exponent was negative, take the reciprocal of 4:
   \[
   8^{-\frac{2}{3}} = \frac{1}{4}
   \]

So, \( 8^{-\frac{2}{3}} = \frac{1}{4} \).

Would you like more details or have any questions about this? Here are some related problems to consider:

1. Simplify \( 27^{-\frac{2}{3}} \).
2. Calculate \( 16^{\frac{3}{4}} \).
3. What is \( 32^{-\frac{4}{5}} \)?
4. Evaluate \( 64^{\frac{2}{3}} \).
5. Simplify \( 125^{-\frac{1}{3}} \).

**Tip**: Remember, when dealing with fractional exponents, the denominator represents the root, and the numerator represents the power.

Learn how to simplify 8^(-2/3) step by step by using the rules of exponents and roots. This example breaks down exponentiation with a negative fractional exponent, the cube root of 8, and the use of reciprocals. Suitable for Grades 8-10.

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