To solve the equation
\[
0 = 2\left(\frac{x}{x+2}\right)^{1/3} + \left(\frac{x+2}{x}\right)^{2/3},
\]
we’ll proceed by breaking it down step-by-step.

### Step 1: Simplify the Equation
First, let’s make substitutions to simplify the expressions:
Let \( u = \left(\frac{x}{x+2}\right)^{1/3} \). Then \(\left(\frac{x}{x+2}\right) = u^3\), and consequently:
\[
\frac{x + 2}{x} = \frac{1}{\frac{x}{x+2}} = \frac{1}{u^3}.
\]

Substituting this into the original equation, we get:
\[
0 = 2u + \left(\frac{1}{u^3}\right)^{2/3}.
\]

### Step 2: Simplify Further
Rewrite \(\left(\frac{1}{u^3}\right)^{2/3}\):
\[
\left(\frac{1}{u^3}\right)^{2/3} = \frac{1^{2/3}}{(u^3)^{2/3}} = \frac{1}{u^2}.
\]

So now the equation is:
\[
0 = 2u + \frac{1}{u^2}.
\]

### Step 3: Combine Terms
Multiply the entire equation by \( u^2 \) to eliminate the denominator:
\[
0 = 2u^3 + 1.
\]

Rearrange this equation:
\[
2u^3 = -1.
\]

### Step 4: Solve for \( u \)
Now divide both sides by 2:
\[
u^3 = -\frac{1}{2}.
\]

Take the cube root of both sides to find \( u \):
\[
u = -\sqrt[3]{\frac{1}{2}} = -\frac{1}{\sqrt[3]{2}}.
\]

### Step 5: Back-Substitute to Solve for \( x \)
Since \( u = \left(\frac{x}{x+2}\right)^{1/3} \), we have:
\[
\left(\frac{x}{x+2}\right)^{1/3} = -\frac{1}{\sqrt[3]{2}}.
\]

Now, cube both sides:
\[
\frac{x}{x+2} = -\frac{1}{2}.
\]

Then, multiply both sides by \( x + 2 \):
\[
x = -\frac{1}{2}(x + 2).
\]

Expanding and solving for \( x \):
\[
x = -\frac{1}{2}x - 1.
\]

Add \(\frac{1}{2}x\) to both sides:
\[
\frac{3}{2}x = -1.
\]

Now, multiply both sides by \(\frac{2}{3}\):
\[
x = -\frac{2}{3}.
\]

### Solution
Thus, the solution to the equation is:
\[
x = -\frac{2}{3}.
\]

Would you like more details or have any questions about any step?

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Here are 5 questions for further exploration:
1. How does the process of substituting variables simplify complex expressions in equations?
2. What alternative methods could solve this type of equation?
3. How does one approach cube roots in real numbers, especially with negative values?
4. Why is it valid to multiply both sides of an equation by \(u^2\)?
5. How does the choice of substitution affect the solvability of equations?

**Tip:** For equations involving fractional exponents, substitution often simplifies the expressions, making the equation more manageable.

Explore the solution to a complex algebraic equation involving rational exponents: 0 = 2(x/x+2)^(1/3) + (x+2/x)^(2/3). This solution uses substitution and exponentiation to simplify and solve the equation, making it accessible for students in Grades 11-12.

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