https://file.aiquickdraw.com/mathbot/file/1725599106934wqqwu4x9.jpg

Find solution and identify any extraneous solution if they exist

The problem extracted from the image reads:

\[
\sqrt[3]{2x - 1} = 2
\]

To solve this equation, follow these steps:

1. **Cube both sides** to eliminate the cube root:
   \[
   \left(\sqrt[3]{2x - 1}\right)^3 = 2^3
   \]
   Simplifying both sides:
   \[
   2x - 1 = 8
   \]

2. **Solve for \(x\)**:
   \[
   2x = 8 + 1
   \]
   \[
   2x = 9
   \]
   \[
   x = \frac{9}{2}
   \]

Thus, the solution is \( x = \frac{9}{2} \).

### Checking for Extraneous Solutions
An extraneous solution arises when the original equation has restrictions that are not satisfied by the obtained solution. In this case, since cubing is a reversible operation for real numbers and there are no domain restrictions from the cube root, \( x = \frac{9}{2} \) is a valid solution without any extraneous solutions.

Do you need further details or have any questions?

Here are 5 related questions:
1. What happens if we square both sides of a square root equation instead of cubing?
2. How do you determine if a solution is extraneous in equations with radicals?
3. What is the general process to solve an equation involving a higher-degree root?
4. How would the solution change if the equation had a fourth root instead of a cube root?
5. Can there be multiple solutions to cube root equations?

**Tip**: Always check for extraneous solutions, especially when dealing with square or higher roots, as operations may introduce invalid solutions!

Find the solution of the equation ∛(2x - 1) = 2.

This page explains how to solve the cube root equation ∛(2x - 1) = 2 by cubing both sides and solving for x. Learn the steps to arrive at the solution x = 9/2, and check for extraneous solutions. Suitable for high school algebra students.

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