The value of `(x^((1)/(3))+x^(-(1)/(3)))(x^((2)/(3))-1+x^(-(2)/(3)))` is

Options

A. `x^(1)+x^((2)/(3))`
B. `x+x^(-(1)/(3))`
C. `x^((1)/(3))+x^(-1)`
D. `x+x^(-1)`

Answer

To solve the expression \((x^{\frac{1}{3}} + x^{-\frac{1}{3}})(x^{\frac{2}{3}} - 1 + x^{-\frac{2}{3}})\), we can 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 \(B\)
Notice that \(B\) can be rewritten using the identity for the sum and difference of cubes:
\[ B = x^{\frac{2}{3}} - 1 + x^{-\frac{2}{3}} = (x^{\frac{2}{3}} + x^{-\frac{2}{3}}) - 1 \]

### Step 3: Simplify \(B\)
We can express \(x^{\frac{2}{3}} + x^{-\frac{2}{3}}\) in terms of \(A\):
\[ x^{\frac{2}{3}} + x^{-\frac{2}{3}} = (x^{\frac{1}{3}})^2 + (x^{-\frac{1}{3}})^2 = (x^{\frac{1}{3}} + x^{-\frac{1}{3}})^2 - 2(x^{\frac{1}{3}})(x^{-\frac{1}{3}}) \]
Thus,
\[ B = A^2 - 2 - 1 = A^2 - 3 \]

### Step 4: Substitute \(B\) back into the expression
Now substitute \(B\) back into the original expression:
\[ A \cdot B = (x^{\frac{1}{3}} + x^{-\frac{1}{3}})(A^2 - 3) \]

### Step 5: Expand the expression
Expanding this gives:
\[ = A^3 - 3A \]

### Step 6: Substitute back \(A\)
Now substitute \(A = x^{\frac{1}{3}} + x^{-\frac{1}{3}}\):
\[ = (x^{\frac{1}{3}} + x^{-\frac{1}{3}})^3 - 3(x^{\frac{1}{3}} + x^{-\frac{1}{3}}) \]

### Step 7: Final expression
The final expression can be simplified further if needed, but the value of the original expression is:
\[ = (x^{\frac{1}{3}} + x^{-\frac{1}{3}})^3 - 3(x^{\frac{1}{3}} + x^{-\frac{1}{3}}) \]