Simplify :
` (x^((1)/(3)) - x^(-(1)/(3)))(x^((2)/(3)) +1+x^(-(2)/(3)))`

Answer

To simplify the expression \( (x^{\frac{1}{3}} - x^{-\frac{1}{3}})(x^{\frac{2}{3}} + 1 + x^{-\frac{2}{3}}) \), we will follow these steps:

### Step 1: Identify the components of the expression
We have two parts in the expression:
1. \( A = x^{\frac{1}{3}} - x^{-\frac{1}{3}} \)
2. \( B = x^{\frac{2}{3}} + 1 + x^{-\frac{2}{3}} \)

### Step 2: Rewrite the second part \( B \)
We can rewrite \( B \) as:
\[ B = x^{\frac{2}{3}} + 1 + x^{-\frac{2}{3}} = x^{\frac{2}{3}} + 1 + \frac{1}{x^{\frac{2}{3}}} \]

### Step 3: Recognize the structure of \( B \)
Notice that \( B \) can be expressed as:
\[ B = (x^{\frac{2}{3}} + \frac{1}{x^{\frac{2}{3}}}) + 1 \]
Let \( y = x^{\frac{1}{3}} \). Then \( y^2 = x^{\frac{2}{3}} \) and \( \frac{1}{y^2} = x^{-\frac{2}{3}} \). Thus,
\[ B = y^2 + 1 + \frac{1}{y^2} \]

### Step 4: Simplify \( B \)
Using the identity \( a^2 + b^2 + 1 = (a+b)^2 - ab \), we can rewrite \( B \):
\[ B = (y + \frac{1}{y})^2 - 2 \]
where \( a = y \) and \( b = \frac{1}{y} \).

### Step 5: Expand the product \( A \cdot B \)
Now we multiply \( A \) and \( B \):
\[ A \cdot B = (y - \frac{1}{y})((y + \frac{1}{y})^2 - 2) \]

### Step 6: Expand further
Using the difference of squares:
\[ A \cdot B = (y - \frac{1}{y})(y^2 + 2 + \frac{1}{y^2} - 2) = (y - \frac{1}{y})(y^2 + \frac{1}{y^2}) \]

### Step 7: Use the identity for cubes
Recognize that:
\[ y^3 - \frac{1}{y^3} = (y - \frac{1}{y})\left((y^2 + \frac{1}{y^2}) + 1\right) \]
Thus,
\[ A \cdot B = y^3 - \frac{1}{y^3} \]

### Step 8: Substitute back for \( y \)
Substituting back \( y = x^{\frac{1}{3}} \):
\[ y^3 - \frac{1}{y^3} = x - \frac{1}{x} \]

### Final Answer
Therefore, the simplified expression is:
\[ \boxed{x - \frac{1}{x}} \]