Solve the equation with rational exponents.
left parenthesis x minus 1 right parenthesis Superscript 2 divided by 3 Baseline equals 36

We are given the equation:

\[
(x - 1)^{\frac{2}{3}} = 36
\]

To solve this, follow these steps:

### Step 1: Isolate the term with the exponent
The term with the exponent is already isolated on the left side of the equation:

\[
(x - 1)^{\frac{2}{3}} = 36
\]

### Step 2: Raise both sides of the equation to the reciprocal of the exponent
To eliminate the fractional exponent \(\frac{2}{3}\), raise both sides of the equation to the power of \(\frac{3}{2}\):

\[
\left[ (x - 1)^{\frac{2}{3}} \right]^{\frac{3}{2}} = 36^{\frac{3}{2}}
\]

On the left-hand side, the exponents will cancel each other out:

\[
x - 1 = 36^{\frac{3}{2}}
\]

### Step 3: Simplify \(36^{\frac{3}{2}}\)
To compute \(36^{\frac{3}{2}}\), break it down as follows:

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

So, we have:

\[
x - 1 = 216
\]

### Step 4: Solve for \(x\)
Add 1 to both sides to solve for \(x\):

\[
x = 216 + 1 = 217
\]

### Final Solution:
\[
x = 217
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you simplify expressions with fractional exponents?
2. What happens when you raise both sides of an equation to a power?
3. How can you solve equations involving square roots?
4. What are the steps to solve an equation with exponents greater than 1?
5. How do you check the solution to an equation involving exponents?

**Tip:** When dealing with fractional exponents, remember that the denominator represents the root, and the numerator represents the power!

Solve the equation with rational exponents: (x - 1)^(2/3) = 36

Learn how to solve the equation (x - 1)^(2/3) = 36 by raising both sides to the reciprocal exponent. This problem covers rational exponents and exponentiation, providing a detailed solution for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation