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.