Let's solve the given equation step by step. The equation provided is:

$\frac{(x - 1)^{2/3}}{3} + \frac{(x - 1)^{1/3}}{3} - 2 = 0$

First, let's multiply the entire equation by 3 to eliminate the fractions:

$(x - 1)^{2/3} + (x - 1)^{1/3} - 6 = 0$

Next, let's introduce a substitution to simplify the equation. Let:

$y = (x - 1)^{1/3}$

So, the equation becomes:

$y^2 + y - 6 = 0$

Now, we solve the quadratic equation $y^2 + y - 6 = 0$. To do that, we can factor it:

$y^2 + y - 6 = (y - 2)(y + 3) = 0$

Setting each factor equal to zero gives:

$y - 2 = 0 \quad \text{or} \quad y + 3 = 0$

So,

$y = 2 \quad \text{or} \quad y = -3$

Now, recall that $y = (x - 1)^{1/3}$. So we have two cases to consider:

Case 1: $y = 2$

$(x - 1)^{1/3} = 2$

Cubing both sides:

$x - 1 = 8$

$x = 9$

Case 2: $y = -3$

$(x - 1)^{1/3} = -3$

Cubing both sides:

$x - 1 = -27$

$x = -26$

Thus, the solution set is:

$\{9, -26\}$

If you have any questions or need further details, feel free to ask!

Here are 5 related questions:

1. How do you solve cubic equations involving radicals?
2. What is the method to solve higher-degree polynomial equations?
3. Can you explain how to simplify expressions involving fractional exponents?
4. What are the key steps to factor quadratic equations?
5. How do you solve equations that involve complex numbers?

Tip: When solving equations, substituting variables can simplify the process, especially when dealing with exponents and radicals.