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
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To solve the equation \( (x - 2)^{\frac{2}{3}} = 36 \), follow these steps:

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

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

Simplifying the left side:

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

### Step 2: Solve \(36^{\frac{3}{2}}\).
First, rewrite \(36^{\frac{3}{2}}\) as:

\[
36^{\frac{3}{2}} = (36^{\frac{1}{2}})^3 = (6)^3 = 216
\]

Thus, the equation becomes:

\[
x - 2 = 216
\]

### Step 3: Solve for \(x\).
Add 2 to both sides:

\[
x = 216 + 2 = 218
\]

Thus, the solution is \(x = 218\).

### Final Answer:
\[
x = 218
\]

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Do you need further explanation or more details on any part of the process? Here are five related questions:

1. How do you simplify expressions with fractional exponents?
2. What is the general method to solve equations with rational exponents?
3. Can you explain why raising both sides to the reciprocal power works?
4. How do you handle equations where the base has a negative exponent?
5. What are some common pitfalls when working with fractional exponents?

**Tip:** When working with fractional exponents, it’s often helpful to rewrite them as roots and powers for better clarity.

Solve the equation with rational exponents: (x - 2)^(2/3) = 36

Learn how to solve the equation (x - 2)^(2/3) = 36 using rational exponents. This step-by-step guide covers key algebraic concepts, including exponentiation and radical operations. Suitable for students in Grades 9-11.

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