If `x = 2^(1//3) + 2^(-1//3)`, then the value of `2x^3` is:

Options

A. `6x + 5`
B. `5x + 6`
C. `6x -5`
D. `5x -6`

Answer

To solve the problem where \( x = 2^{1/3} + 2^{-1/3} \) and we need to find the value of \( 2x^3 \), we will follow these steps:

### Step 1: Define \( x \)
Let:
\[ x = 2^{1/3} + 2^{-1/3} \]

### Step 2: Cube \( x \)
To find \( x^3 \), we can use the identity for the cube of a sum:
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]
Here, let \( a = 2^{1/3} \) and \( b = 2^{-1/3} \). Thus,
\[ x^3 = (2^{1/3} + 2^{-1/3})^3 = (2^{1/3})^3 + (2^{-1/3})^3 + 3(2^{1/3})(2^{-1/3})(2^{1/3} + 2^{-1/3}) \]

### Step 3: Calculate \( a^3 \) and \( b^3 \)
Calculating \( a^3 \) and \( b^3 \):
\[ (2^{1/3})^3 = 2 \quad \text{and} \quad (2^{-1/3})^3 = \frac{1}{2} \]
Thus,
\[ a^3 + b^3 = 2 + \frac{1}{2} = \frac{5}{2} \]

### Step 4: Calculate \( ab \)
Next, we calculate \( ab \):
\[ ab = 2^{1/3} \cdot 2^{-1/3} = 2^{(1/3 - 1/3)} = 2^0 = 1 \]

### Step 5: Substitute back into the cube formula
Now substituting back into the cube formula:
\[ x^3 = \frac{5}{2} + 3(1)(x) = \frac{5}{2} + 3x \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ x^3 - 3x - \frac{5}{2} = 0 \]

### Step 7: Multiply the entire equation by 2
To eliminate the fraction, multiply the entire equation by 2:
\[ 2x^3 - 6x - 5 = 0 \]

### Step 8: Solve for \( 2x^3 \)
From the equation, we can express \( 2x^3 \):
\[ 2x^3 = 6x + 5 \]

### Conclusion
Thus, the value of \( 2x^3 \) is:
\[ \boxed{6x + 5} \]