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 can follow these steps:

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

### Step 2: Cube both sides
To find \( x^3 \), we will cube both sides:
\[ x^3 = (2^{1/3} + 2^{-1/3})^3 \]

### Step 3: Apply the formula for the cube of a binomial
Using the formula \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), where \( a = 2^{1/3} \) and \( b = 2^{-1/3} \):
\[ x^3 = (2^{1/3})^3 + (2^{-1/3})^3 + 3(2^{1/3})(2^{-1/3})(2^{1/3} + 2^{-1/3}) \]

### Step 4: Simplify each term
Calculating each term:
- \( (2^{1/3})^3 = 2 \)
- \( (2^{-1/3})^3 = \frac{1}{2} \)
- \( 2^{1/3} \cdot 2^{-1/3} = 1 \)

So, we have:
\[ x^3 = 2 + \frac{1}{2} + 3(1)(x) \]

### Step 5: Combine the terms
Now combine the terms:
\[ x^3 = 2 + 0.5 + 3x \]
\[ x^3 = 2.5 + 3x \]
Or in fraction form:
\[ x^3 = \frac{5}{2} + 3x \]

### Step 6: Multiply by 2 to find \( 2x^3 \)
Now, we want to find \( 2x^3 \):
\[ 2x^3 = 2\left(\frac{5}{2} + 3x\right) \]
Distributing the 2:
\[ 2x^3 = 5 + 6x \]

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