To solve the equation:

\[
\left(x-1\right)^{\frac{2}{3}} = 9
\]

we can proceed by following these steps:

### Step 1: Remove the Fractional Exponent
To eliminate the exponent $ \frac{2}{3} $, raise both sides of the equation to the power of $ \frac{3}{2} $, which is the reciprocal of $ \frac{2}{3} $:

\[
\left( \left(x - 1\right)^{\frac{2}{3}} \right)^{\frac{3}{2}} = 9^{\frac{3}{2}}
\]

This simplifies to:

\[
x - 1 = 9^{\frac{3}{2}}
\]

### Step 2: Calculate $ 9^{\frac{3}{2}} $
To compute $ 9^{\frac{3}{2}} $, we can rewrite it as $ \sqrt{9}^3 $:

\[
9^{\frac{3}{2}} = \left(\sqrt{9}\right)^3 = 3^3 = 27
\]

So now we have:

\[
x - 1 = 27
\]

### Step 3: Solve for $ x $
Add 1 to both sides to solve for $ x $:

\[
x = 27 + 1
\]

\[
x = 28
\]

### Final Answer:
\[
x = 28
\]

Would you like more details, or do you have any questions?

---

Here are five related questions to consider:

1. What if the equation was $ \left(x-1\right)^{\frac{3}{2}} = 9 $?
2. How would the solution change if the exponent was $ \frac{2}{5} $ instead of $ \frac{2}{3} $?
3. What happens if the constant on the right side is negative?
4. How can this method be extended to solve equations with higher fractional powers?
5. How would we solve if the equation was $ \left(2x+3\right)^{\frac{2}{3}} = 16 $?

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

The query submitted by the user: $$ (x-1)^{rac{2}{3}} = 9 $$

Learn how to solve the equation (x-1)^(2/3) = 9 step-by-step. This problem involves fractional exponents and exponentiation rules, and is suitable for students in Grades 8-10. The solution includes simplifying fractional exponents and applying the correct properties of exponents.

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