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The equation given is:

\[
(x - 2)^{\frac{5}{2}} = 243
\]

We will solve this step by step:

### Step 1: Isolate the base with the exponent
The equation is already isolated:
\[
(x - 2)^{\frac{5}{2}} = 243
\]

### Step 2: Remove the rational exponent
To remove the rational exponent \(\frac{5}{2}\), raise both sides of the equation to the reciprocal power, which is \(\frac{2}{5}\):

\[
\left( (x - 2)^{\frac{5}{2}} \right)^{\frac{2}{5}} = 243^{\frac{2}{5}}
\]

This simplifies the left side to:
\[
x - 2 = 243^{\frac{2}{5}}
\]

### Step 3: Simplify \(243^{\frac{2}{5}}\)
To simplify this, note that \(243 = 3^5\), so:
\[
243^{\frac{2}{5}} = (3^5)^{\frac{2}{5}} = 3^2 = 9
\]

Thus:
\[
x - 2 = 9
\]

### Step 4: Solve for \(x\)
Now, solve for \(x\):
\[
x = 9 + 2 = 11
\]

### Final Answer:
The solution set is \(\{ 11 \}\).

Would you like any additional details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you solve equations with other types of rational exponents?
2. What are some common mistakes when solving rational exponent equations?
3. Can there be multiple solutions when solving rational exponent equations?
4. What if the exponent was negative—how would that change the solution?
5. How do rational exponents relate to radicals (roots)?

**Tip:** Always check for extraneous solutions when solving equations involving rational exponents by substituting the solution back into the original equation.

Solve the equation with rational exponents: (x - 2)^(5/2) = 243

Learn how to solve the equation (x - 2)^(5/2) = 243 step-by-step. This algebra problem focuses on rational exponents and involves simplifying fractional powers. Suitable for students in Grades 10-12.

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