https://file.aiquickdraw.com/mathbot/file/17267698944525hjyzpa7.jpg

The equation shown in the image is:

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

To solve this, we can follow these steps:

### Step 1: Cube both sides
Since we have a cube root on the left-hand side, we can remove it by cubing both sides of the equation:

\[
x^2 - 1 = 2^3
\]

\[
x^2 - 1 = 8
\]

### Step 2: Isolate \(x^2\)
Now, add 1 to both sides to isolate \(x^2\):

\[
x^2 = 8 + 1
\]

\[
x^2 = 9
\]

### Step 3: Solve for \(x\)
Take the square root of both sides:

\[
x = \pm 3
\]

So the solutions are:

\[
x = 3 \quad \text{or} \quad x = -3
\]

### Final Answer:
The solutions to the equation are \(x = 3\) and \(x = -3\).

Would you like further details on any part of the solution?

Here are some related questions to explore:
1. What happens if the equation had a fourth root instead of a cube root?
2. Can we have complex solutions for cube-root equations?
3. What would the equation look like if \(x^2 - 1 = 0\)?
4. How can we verify the solutions to this equation?
5. How do cube root functions behave graphically?

**Tip:** Always check your solutions by substituting them back into the original equation!

Solve the equation: cube root of (x^2 - 1) = 2.

This tutorial explains how to solve the cube root equation \( \sqrt[3]{x^2 - 1} = 2 \) using algebraic techniques. Learn how to isolate the quadratic term, square both sides, and solve for \(x\). Perfect for students in Grades 9-11.

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