If `a^(1//3)+b^(1//3)+c^(1//3)=0`, then :

Options

A. `a+b+c=0`
B. `a+b+c=3abc`
C. `a^(3)+b^(3)+c^(3)=0`
D. `(a+b+c)^(3)=27abc`

Answer

To solve the equation \( a^{1/3} + b^{1/3} + c^{1/3} = 0 \), we can use the identity for the sum of cubes.

### Step-by-Step Solution:

1. **Let \( x = a^{1/3} \), \( y = b^{1/3} \), and \( z = c^{1/3} \)**:
\[ x + y + z = 0 \]

2. **Use the identity for the sum of cubes**:
The identity states:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) \]
Since \( x + y + z = 0 \), we can simplify this to:
\[ x^3 + y^3 + z^3 = 3xyz \]

3. **Substituting back**:
We have:
\[ a + b + c = 3 \cdot (a^{1/3} \cdot b^{1/3} \cdot c^{1/3}) \]
This means:
\[ a + b + c = 3 \sqrt[3]{abc} \]

4. **Conclusion**:
Therefore, if \( a^{1/3} + b^{1/3} + c^{1/3} = 0 \), then:
\[ a + b + c = 3 \sqrt[3]{abc} \]