Solve each of the following equatins :
`x^(2//3)+x^(1//3)-2=0`

Answer

To solve the equation \( x^{2/3} + x^{1/3} - 2 = 0 \), we will use a substitution method. Let's go through the steps:

### Step 1: Substitute \( t = x^{1/3} \)
We can let \( t = x^{1/3} \). Therefore, \( t^2 = (x^{1/3})^2 = x^{2/3} \).

Now, we can rewrite the original equation:
\[ t^2 + t - 2 = 0 \]

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

The numbers \(2\) and \(-1\) satisfy these conditions:
\[ (t + 2)(t - 1) = 0 \]

### Step 3: Set each factor to zero
Now, we set each factor equal to zero:
1. \( t + 2 = 0 \)
2. \( t - 1 = 0 \)

From these, we can solve for \(t\):
1. \( t = -2 \)
2. \( t = 1 \)

### Step 4: Substitute back to find \(x\)
Now, we substitute back to find \(x\) using \( t = x^{1/3} \).

1. For \( t = -2 \):
\[ x^{1/3} = -2 \implies x = (-2)^3 = -8 \]

2. For \( t = 1 \):
\[ x^{1/3} = 1 \implies x = 1^3 = 1 \]

### Step 5: Write the final solutions
The solutions to the equation \( x^{2/3} + x^{1/3} - 2 = 0 \) are:
\[ x = -8 \quad \text{and} \quad x = 1 \]