https://file.aiquickdraw.com/mathbot/file/172702051297443fxiyca.jpg

The equation from the image is:

\[
(x - 2)^{\frac{2}{3}} + (x - 2)^{\frac{1}{3}} - 20 = 0
\]

Let's solve this step by step.

### Step 1: Substitution
Let’s set \( y = (x - 2)^{\frac{1}{3}} \). This means that:

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

Now substitute \( y \) and \( y^2 \) into the original equation:

\[
y^2 + y - 20 = 0
\]

### Step 2: Solve the quadratic equation
We now solve the quadratic equation:

\[
y^2 + y - 20 = 0
\]

Using the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = 1 \), and \( c = -20 \).

Substitute these values into the formula:

\[
y = \frac{-1 \pm \sqrt{1^2 - 4(1)(-20)}}{2(1)}
\]
\[
y = \frac{-1 \pm \sqrt{1 + 80}}{2}
\]
\[
y = \frac{-1 \pm \sqrt{81}}{2}
\]
\[
y = \frac{-1 \pm 9}{2}
\]

So, the two possible values for \( y \) are:

\[
y_1 = \frac{-1 + 9}{2} = 4 \quad \text{and} \quad y_2 = \frac{-1 - 9}{2} = -5
\]

### Step 3: Back-substitute to find \( x \)
We now return to our substitution \( y = (x - 2)^{\frac{1}{3}} \).

For \( y_1 = 4 \):

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

Cubing both sides:

\[
64 = x - 2
\]

Thus,

\[
x = 64 + 2 = 66
\]

For \( y_2 = -5 \):

\[
-5 = (x - 2)^{\frac{1}{3}}
\]

Cubing both sides:

\[
-125 = x - 2
\]

Thus,

\[
x = -125 + 2 = -123
\]

### Final Answer:
The solutions to the equation are:

\[
x = 66 \quad \text{and} \quad x = -123
\]

Would you like any further details or clarifications? 

Here are some related questions:
1. What is the general process for solving equations with fractional exponents?
2. How does substitution simplify solving complex equations?
3. Can all equations with fractional powers be solved by substitution?
4. What would happen if there were multiple terms with different powers of \( (x - 2) \)?
5. How does the quadratic formula help in solving non-linear equations?

**Tip:** When dealing with fractional exponents, substitution is a great technique to reduce the complexity of the equation.

Learn how to solve the equation (x - 2)^(2/3) + (x - 2)^(1/3) - 20 = 0 step-by-step using substitution and the quadratic formula. This solution covers important algebraic techniques and concepts such as fractional exponents and the quadratic formula. Suitable for Grades 10-12 students.

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