To solve \((8x^6)^{-\frac{2}{3}}\), follow these steps:

1. **Apply the Negative Exponent Rule**:  
   \((a^m)^{-n} = \frac{1}{(a^m)^n}\).  
   So,  
   \[
   (8x^6)^{-\frac{2}{3}} = \frac{1}{(8x^6)^{\frac{2}{3}}}
   \]

2. **Simplify the Expression Inside the Fraction**:  
   Break it down into two separate parts:
   \[
   \frac{1}{8^{\frac{2}{3}} \cdot (x^6)^{\frac{2}{3}}}
   \]

3. **Simplify \(8^{\frac{2}{3}}\)**:  
   \(8^{\frac{2}{3}}\) can be simplified by first taking the cube root of 8, which is 2, and then squaring the result:
   \[
   8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4
   \]

4. **Simplify \((x^6)^{\frac{2}{3}}\)**:  
   Use the power rule \((a^m)^n = a^{mn}\) to simplify:
   \[
   (x^6)^{\frac{2}{3}} = x^{6 \cdot \frac{2}{3}} = x^4
   \]

5. **Combine the Results**:
   \[
   \frac{1}{4x^4}
   \]

So, the final answer is:
\[
\boxed{\frac{1}{4x^4}}
\]

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:
1. How would you solve \((27y^9)^{\frac{1}{3}}\)?
2. What is the result of simplifying \((16a^4)^{-\frac{1}{2}}\)?
3. How do you simplify \((2x^2)^{\frac{3}{4}}\)?
4. Can you explain how to expand \((x^3 + 2)^2\)?
5. What does \((x^5y^{-2})^{\frac{2}{5}}\) simplify to?

**Tip:** When dealing with fractional exponents, remember that the numerator represents a power, while the denominator represents a root.

Learn how to solve the exponential expression (8x^6)^{-2/3} step-by-step. This problem involves applying the negative exponent rule and simplifying fractional exponents. Suitable for advanced high school students.

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