If `x =3 ^(1//3) + 3 ^(- 1//3)` , then ` 3x ^(3)- 10` is equal to

Options

A. `- 3x`
B. `3x`
C. `-9x`
D. 9x

Answer

To solve the problem, we need to find the value of \(3x^3 - 10\) given that \(x = 3^{1/3} + 3^{-1/3}\).

### Step-by-Step Solution:

1. **Identify the expression for \(x\)**:
\[ x = 3^{1/3} + 3^{-1/3} \]

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

3. **Calculate \(a^3\) and \(b^3\)**:
\[ (3^{1/3})^3 = 3 \quad \text{and} \quad (3^{-1/3})^3 = \frac{1}{3} \]
Therefore,
\[ a^3 + b^3 = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \]

4. **Calculate \(3ab\)**:
\[ ab = 3^{1/3} \cdot 3^{-1/3} = 1 \quad \Rightarrow \quad 3ab = 3 \cdot 1 = 3 \]

5. **Combine the results**:
\[ x^3 = \frac{10}{3} + 3(1)(x) = \frac{10}{3} + 3x \]

6. **Multiply by 3**:
\[ 3x^3 = 3\left(\frac{10}{3} + 3x\right) = 10 + 9x \]

7. **Subtract 10**:
\[ 3x^3 - 10 = 10 + 9x - 10 = 9x \]

### Final Answer:
\[ 3x^3 - 10 = 9x \]