If `p=x^(1//3)+x^(-1//3)` then `p^3-3p` is equal to :

Options

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

Answer

To solve the problem where \( p = x^{1/3} + x^{-1/3} \), we need to find the value of \( p^3 - 3p \).

### Step-by-Step Solution:

1. **Define \( p \)**:
\[ p = x^{1/3} + x^{-1/3} \]

2. **Cube \( p \)**:
Using the identity for the cube of a sum, \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), we can write:
\[ p^3 = (x^{1/3} + x^{-1/3})^3 = (x^{1/3})^3 + (x^{-1/3})^3 + 3(x^{1/3})(x^{-1/3})(x^{1/3} + x^{-1/3}) \]

3. **Calculate \( (x^{1/3})^3 \) and \( (x^{-1/3})^3 \)**:
\[ (x^{1/3})^3 = x \quad \text{and} \quad (x^{-1/3})^3 = \frac{1}{x} \]
Therefore,
\[ p^3 = x + \frac{1}{x} + 3(x^{1/3})(x^{-1/3})(p) \]

4. **Calculate \( (x^{1/3})(x^{-1/3}) \)**:
\[ (x^{1/3})(x^{-1/3}) = 1 \]
So, we can simplify \( p^3 \):
\[ p^3 = x + \frac{1}{x} + 3p \]

5. **Rearranging the equation**:
Now, we want to find \( p^3 - 3p \):
\[ p^3 - 3p = x + \frac{1}{x} \]

6. **Final Result**:
Therefore, we conclude that:
\[ p^3 - 3p = x + \frac{1}{x} \]