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The equation to solve is:

\[
\left( x^2 - x - 12 \right)^{3/2} = 27
\]

Let's solve this step by step.

### Step 1: Isolate the term with the exponent
We start by eliminating the fractional exponent. To do this, we raise both sides of the equation to the power of \( \frac{2}{3} \).

\[
\left( (x^2 - x - 12)^{3/2} \right)^{2/3} = 27^{2/3}
\]

The left side simplifies to \( x^2 - x - 12 \), and the right side becomes \( 27^{2/3} = 9 \).

Now we have the equation:

\[
x^2 - x - 12 = 9
\]

### Step 2: Simplify the equation
Subtract 9 from both sides:

\[
x^2 - x - 12 - 9 = 0
\]

This simplifies to:

\[
x^2 - x - 21 = 0
\]

### Step 3: Solve the quadratic equation
We now solve the quadratic equation:

\[
x^2 - x - 21 = 0
\]

We can solve this using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-21)}}{2(1)}
\]

Simplifying:

\[
x = \frac{1 \pm \sqrt{1 + 84}}{2}
\]
\[
x = \frac{1 \pm \sqrt{85}}{2}
\]

Thus, the solutions are:

\[
x = \frac{1 + \sqrt{85}}{2}, \quad x = \frac{1 - \sqrt{85}}{2}
\]

### Approximate values:
- \( x_1 \approx \frac{1 + 9.22}{2} = 5.11 \)
- \( x_2 \approx \frac{1 - 9.22}{2} = -4.11 \)

### Final Answer:
The solutions to the equation are approximately:
\[
x \approx 5.11, -4.11
\]

Would you like a more detailed explanation or help with any other step?

Here are five related questions:
1. How is the quadratic formula derived?
2. How would the solution change if the exponent were different (e.g., \(5/2\) instead of \(3/2\))?
3. How do you verify solutions to equations like this one?
4. What is the geometric interpretation of solving a quadratic equation?
5. What are other methods to solve quadratic equations?

**Tip:** When using the quadratic formula, always simplify the discriminant first to avoid calculation errors!

Learn how to solve the equation (x^2 - x - 12)^(3/2) = 27 with a detailed step-by-step explanation. This problem involves exponents and solving quadratic equations. Suitable for high school students in Grades 9-12.

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