Solve the equation with rational exponents.
left parenthesis x minus 2 right parenthesis Superscript 2 divided by 3 Baseline equals 36
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is

We are given the equation:

\[
(x - 2)^{\frac{2}{3}} = 36
\]

To solve this, we need to isolate \(x\). Here's the step-by-step solution:

### Step 1: Eliminate the fractional exponent
To remove the \(\frac{2}{3}\)-exponent, raise both sides of the equation to the power of \(\frac{3}{2}\), which is the reciprocal of \(\frac{2}{3}\).

\[
\left[(x - 2)^{\frac{2}{3}}\right]^{\frac{3}{2}} = 36^{\frac{3}{2}}
\]

This simplifies to:

\[
x - 2 = 36^{\frac{3}{2}}
\]

### Step 2: Simplify \(36^{\frac{3}{2}}\)
To calculate \(36^{\frac{3}{2}}\), recall that:

\[
36^{\frac{3}{2}} = \left( \sqrt{36} \right)^3 = 6^3 = 216
\]

So we have:

\[
x - 2 = 216
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\) by adding 2 to both sides:

\[
x = 216 + 2 = 218
\]

### Final Solution:
The solution is:

\[
x = 218
\]

### Conclusion:
Thus, the solution set is \( \{ 218 \} \).

---

Would you like further details or clarification? Here are five related questions for further practice:

1. Solve \( (x + 3)^{\frac{3}{4}} = 64 \).
2. Solve \( (x - 5)^{\frac{2}{5}} = 32 \).
3. Solve \( (x + 1)^{\frac{1}{2}} = 49 \).
4. Solve \( (x + 4)^{\frac{4}{3}} = 81 \).
5. Solve \( (x - 6)^{\frac{5}{2}} = 243 \).

**Tip:** When dealing with fractional exponents, always apply the reciprocal exponent to both sides to eliminate the fraction.

Solve the equation with rational exponents: (x - 2)^(2/3) = 36. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Learn how to solve the equation (x - 2)^(2/3) = 36 with a detailed step-by-step explanation. This problem focuses on algebraic manipulation of rational exponents and exponentiation rules. Suitable for students in Grades 9-11.

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