If `x=a^(1//3)-a^(-1//3)," then what is " x^(3) + 3x` equal to ?

Options

A. zero
B. `a+(1/a)`
C. `a-(1/a)`
D. `a^(3) +(1/(a^(3)))`

Explanation

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

### Step-by-Step Solution:

1. **Start with the given expression for \( x \)**:
\[ x = a^{1/3} - a^{-1/3} \]

2. **Cube both sides to find \( x^3 \)**:
\[ x^3 = (a^{1/3} - a^{-1/3})^3 \]

3. **Use the identity for the cube of a binomial**:
The formula for \( (A - B)^3 \) is:
\[ A^3 - B^3 - 3AB(A - B) \]
Here, let \( A = a^{1/3} \) and \( B = a^{-1/3} \).

4. **Calculate \( A^3 \) and \( B^3 \)**:
\[ A^3 = (a^{1/3})^3 = a \]
\[ B^3 = (a^{-1/3})^3 = a^{-1} \]

5. **Calculate \( 3AB \)**:
\[ 3AB = 3(a^{1/3})(a^{-1/3}) = 3(a^{1/3 - 1/3}) = 3(1) = 3 \]

6. **Substitute back into the identity**:
\[ x^3 = a - a^{-1} - 3(a^{1/3})(a^{-1/3})(a^{1/3} - a^{-1/3}) \]
Since \( A - B = x \):
\[ x^3 = a - a^{-1} - 3x \]

7. **Rearranging the equation**:
\[ x^3 + 3x = a - a^{-1} \]

8. **Final result**:
\[ x^3 + 3x = a - \frac{1}{a} \]

### Conclusion:
Thus, the expression \( x^3 + 3x \) is equal to:
\[ \boxed{a - \frac{1}{a}} \]