If `x=2^(1//3)+2^(-1//3)` then the value of `2x^3-6x` will be :

Options

A. 5
B. -5
C. 1
D. 0

Answer

To solve the problem, we need to find the value of \(2x^3 - 6x\) given that \(x = 2^{1/3} + 2^{-1/3}\).

### Step-by-Step Solution:

1. **Define \(x\)**:
\[ x = 2^{1/3} + 2^{-1/3} \]

2. **Cube \(x\)**:
We will use the identity \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\).
Here, let \(a = 2^{1/3}\) and \(b = 2^{-1/3}\).
Thus, we have:
\[ x^3 = (2^{1/3} + 2^{-1/3})^3 = (2^{1/3})^3 + (2^{-1/3})^3 + 3(2^{1/3})(2^{-1/3})(2^{1/3} + 2^{-1/3}) \]

3. **Calculate \(a^3\) and \(b^3\)**:
\[ (2^{1/3})^3 = 2 \quad \text{and} \quad (2^{-1/3})^3 = \frac{1}{2} \]
Therefore:
\[ x^3 = 2 + \frac{1}{2} + 3(2^{1/3})(2^{-1/3})x \]

4. **Calculate \(ab\)**:
\[ ab = 2^{1/3} \cdot 2^{-1/3} = 1 \]
Hence:
\[ x^3 = 2 + \frac{1}{2} + 3(1)x = 2 + \frac{1}{2} + 3x \]

5. **Combine terms**:
\[ x^3 = \frac{5}{2} + 3x \]

6. **Rearrange the equation**:
\[ x^3 - 3x = \frac{5}{2} \]

7. **Multiply the entire equation by 2**:
\[ 2x^3 - 6x = 5 \]

Thus, the value of \(2x^3 - 6x\) is \(5\).

### Final Answer:
\[ \boxed{5} \]