If `x=2^(1//3)-2^(-1//3)` then what is the value of `2x^(3)+6x` ?

Options

A. 1
B. 2
C. 3
D. 4

Explanation

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

### Step 1: Define \(a\)
Let \(a = 2^{1/3}\). Then we can rewrite \(x\) as:
\[ x = a - \frac{1}{a} \]

### Step 2: Calculate \(x^2\)
Now, we will find \(x^2\):
\[ x^2 = \left(a - \frac{1}{a}\right)^2 = a^2 - 2 + \frac{1}{a^2} \]
Using \(a^2 = (2^{1/3})^2 = 2^{2/3}\), we have:
\[ x^2 = 2^{2/3} - 2 + 2^{-2/3} \]

### Step 3: Calculate \(x^3\)
Next, we will find \(x^3\) using the identity:
\[ x^3 = \left(a - \frac{1}{a}\right)^3 = a^3 - 3a\left(\frac{1}{a}\right) + \frac{1}{a^3} \]
Since \(a^3 = 2\) and \(\frac{1}{a^3} = 2^{-1}\):
\[ x^3 = 2 - 3 + \frac{1}{2} = -1 + \frac{1}{2} = -\frac{1}{2} \]

### Step 4: Calculate \(2x^3\)
Now we can calculate \(2x^3\):
\[ 2x^3 = 2 \times -\frac{1}{2} = -1 \]

### Step 5: Calculate \(6x\)
Next, we need to find \(6x\):
\[ 6x = 6\left(a - \frac{1}{a}\right) = 6a - 6\frac{1}{a} \]
Substituting \(a = 2^{1/3}\):
\[ 6x = 6 \times 2^{1/3} - 6 \times 2^{-1/3} \]

### Step 6: Combine \(2x^3 + 6x\)
Now we can combine the results:
\[ 2x^3 + 6x = -1 + 6\left(2^{1/3} - 2^{-1/3}\right) \]

### Final Calculation
To find the final value, we need to evaluate \(6\left(2^{1/3} - 2^{-1/3}\right)\):
\[ 2^{1/3} - 2^{-1/3} = 2^{1/3} - \frac{1}{2^{1/3}} = \frac{2^{2/3} - 1}{2^{1/3}} \]
Thus,
\[ 6x = 6 \cdot \frac{2^{2/3} - 1}{2^{1/3}} = \frac{6(2^{2/3} - 1)}{2^{1/3}} = 6 \cdot 2^{1/3} - 6 \cdot 2^{-1/3} \]

Finally, we can conclude:
\[ 2x^3 + 6x = -1 + 6\left(2^{1/3} - 2^{-1/3}\right) \]

### Answer
The value of \(2x^3 + 6x\) is \( -1 + 6\left(2^{1/3} - 2^{-1/3}\right)\).