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Let's solve the equation \( (x - 2)^{\frac{2}{3}} = 9 \).

### Steps:

1. **Raise both sides to the power of \( \frac{3}{2} \)**:
   \[
   \left( (x - 2)^{\frac{2}{3}} \right)^{\frac{3}{2}} = 9^{\frac{3}{2}}
   \]
   On the left-hand side, the exponents cancel out, leaving:
   \[
   x - 2 = 9^{\frac{3}{2}}
   \]
   
2. **Simplify \( 9^{\frac{3}{2}} \)**:
   \[
   9^{\frac{3}{2}} = \left( \sqrt{9} \right)^3 = 3^3 = 27
   \]
   So the equation becomes:
   \[
   x - 2 = 27
   \]

3. **Solve for \( x \)**:
   \[
   x = 27 + 2 = 29
   \]

Thus, the solution is \( x = 29 \).

So the correct answer is: **A. The solution set is {29}.**

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Do you want more details or have any questions? 

Here are 5 related questions to expand on the information:
1. What happens when the exponent in the equation is a negative fraction?
2. How would the solution change if the base of the power was negative?
3. Can the method used here be applied to equations with higher-order roots?
4. How do you verify the solution for equations involving rational exponents?
5. What if the equation had multiple solutions? How would you check for extraneous roots?

**Tip**: Always simplify powers carefully, especially when dealing with fractional exponents.

Learn how to solve equations with rational exponents like (x - 2)^(2/3) = 9. This example walks through raising both sides to the power of 3/2, simplifying, and solving for x. Suitable for students in Grades 9-12.

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