Solve: `x^((2)/(3))+x^((1)/(3))-2=0`.

Answer

To solve the equation \( x^{\frac{2}{3}} + x^{\frac{1}{3}} - 2 = 0 \), we can follow these steps:

### Step 1: Substitute \( y = x^{\frac{1}{3}} \)
Let \( y = x^{\frac{1}{3}} \). Then, we can express \( x^{\frac{2}{3}} \) in terms of \( y \):
\[ x^{\frac{2}{3}} = (x^{\frac{1}{3}})^2 = y^2 \]
Now, substitute \( y \) into the equation:
\[ y^2 + y - 2 = 0 \]

### Step 2: Factor the quadratic equation
Next, we need to factor the quadratic equation \( y^2 + y - 2 = 0 \). We look for two numbers that multiply to \(-2\) (the constant term) and add to \(1\) (the coefficient of \(y\)):
\[ (y - 1)(y + 2) = 0 \]

### Step 3: Solve for \( y \)
Setting each factor equal to zero gives us:
1. \( y - 1 = 0 \) → \( y = 1 \)
2. \( y + 2 = 0 \) → \( y = -2 \)

### Step 4: Substitute back to find \( x \)
Now we substitute back for \( y \) to find \( x \):
1. For \( y = 1 \):
\[ x^{\frac{1}{3}} = 1 \implies x = 1^3 = 1 \]
2. For \( y = -2 \):
\[ x^{\frac{1}{3}} = -2 \implies x = (-2)^3 = -8 \]

### Step 5: Write the final solutions
The solutions for the original equation are:
\[ x = 1 \quad \text{and} \quad x = -8 \]

### Summary of Solutions
The solutions to the equation \( x^{\frac{2}{3}} + x^{\frac{1}{3}} - 2 = 0 \) are:
\[ \boxed{1 \text{ and } -8} \]