Solve the equation.

​(xminus−11​)Superscript 2 divided by 32/3plus+​(xminus−11​)Superscript 1 divided by 31/3minus−22equals=0

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Part 1

The solution set is

StartSet nothing EndSetenter your response here.

​(Simplify your answer. Type an integer or a fraction. Type exact​ answers, using radicals as needed. Use a comma to separate answers as needed. Express complex numbers in terms of

ii​.)

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.

Solve the equation (x − 1)^(2/3) / 3 + (x − 1)^(1/3) / 3 − 2 = 0

This page provides a step-by-step guide on solving the equation (x − 1)^(2/3) / 3 + (x − 1)^(1/3) / 3 − 2 = 0 using substitution and quadratic solving techniques. Ideal for students in Grades 10-12 looking to strengthen their skills in algebra and radical equations.

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