The equation you've provided can be rewritten as:

$\frac{(x - 11)^2}{3^{2/3}} + \frac{(x - 11)}{3^{1/3}} - 22 = 0$

To solve this equation, we can proceed step by step.

Step 1: Simplify the equation

Let $y = x - 11$. Substituting $y$ into the equation, we have:

$\frac{y^2}{3^{2/3}} + \frac{y}{3^{1/3}} - 22 = 0$

Step 2: Clear the denominators

Multiply the entire equation by $3^{2/3}$ to clear the fractions:

$y^2 + 3^{1/3} \cdot y - 22 \cdot 3^{2/3} = 0$

Step 3: Solve the quadratic equation

This is a quadratic equation in $y$. To solve it, we use the quadratic formula:

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

Here, $a = 1$, $b = 3^{1/3}$, and $c = -22 \cdot 3^{2/3}$. Substituting these values:

$y = \frac{-3^{1/3} \pm \sqrt{(3^{1/3})^2 + 4 \cdot 22 \cdot 3^{2/3}}}{2}$

Simplifying the square root term:

$y = \frac{-3^{1/3} \pm \sqrt{3^{2/3} + 88 \cdot 3^{2/3}}}{2}$

$y = \frac{-3^{1/3} \pm \sqrt{3^{2/3}(1 + 88)}}{2}$

$y = \frac{-3^{1/3} \pm 3^{1/3} \sqrt{89}}{2}$

Factoring out $3^{1/3}$:

$y = \frac{3^{1/3}(-1 \pm \sqrt{89})}{2}$

Step 4: Substitute back to find $x$

Recall that $y = x - 11$. Therefore:

$x = 11 + \frac{3^{1/3}(-1 \pm \sqrt{89})}{2}$

Final Answer

The solutions to the equation are:

$x_1 = 11 + \frac{3^{1/3}(-1 + \sqrt{89})}{2}$ $x_2 = 11 + \frac{3^{1/3}(-1 - \sqrt{89})}{2}$

Would you like further details on any of the steps?

Here are 5 related questions to explore:

1. What are the properties of cube roots used in solving equations?
2. How does the quadratic formula work in solving polynomial equations?
3. Can we factor quadratic equations involving cube roots?
4. How do transformations, like shifting, affect the solutions of equations?
5. What are some common methods for solving equations involving radicals?

Tip: Always check your final solutions by substituting them back into the original equation to verify their correctness.