To solve the equation:

\[
(x^2 - 3x + 3)^{\frac{3}{2}} - 1 = 0
\]

### Step 1: Isolate the term with the exponent
Add 1 to both sides:

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

### Step 2: Eliminate the exponent
Take both sides to the power of \(\frac{2}{3}\) to eliminate the exponent \(\frac{3}{2}\):

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

### Step 3: Solve the quadratic equation
Now, solve the equation:

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

Subtract 1 from both sides:

\[
x^2 - 3x + 2 = 0
\]

### Step 4: Factor the quadratic
Factor the quadratic equation:

\[
(x - 1)(x - 2) = 0
\]

### Step 5: Solve for \(x\)
Set each factor equal to zero:

1. \(x - 1 = 0 \implies x = 1\)
2. \(x - 2 = 0 \implies x = 2\)

### Final Solution:
The solutions are:

\[
x = 1 \quad \text{or} \quad x = 2
\]

Would you like more details on this solution?

Here are five related questions:
1. How can we verify the solutions for this equation?
2. What happens if the exponent is \(\frac{5}{2}\) instead of \(\frac{3}{2}\)?
3. Can this quadratic be solved using the quadratic formula?
4. How does the discriminant help in determining the nature of solutions for quadratic equations?
5. How do fractional exponents work in equations?

**Tip**: Always check if both solutions satisfy the original equation, especially when dealing with powers and exponents.

This guide explains how to solve the equation (x^2 - 3x + 3)^{3/2} - 1 = 0 step-by-step. Explore algebraic concepts such as solving quadratic equations and working with fractional exponents. Suitable for students in Grades 9-11.

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