If `a^(1/3)+b^(1/3)+c^(1/3)=0` then which one of the following options is correct?

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. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the variables
Let:
- \( x = a^{1/3} \)
- \( y = b^{1/3} \)
- \( z = c^{1/3} \)

Given that:
\[ x + y + z = 0 \]

### Step 2: Use the sum of cubes formula
We know from algebra that if \( x + y + z = 0 \), then:
\[ x^3 + y^3 + z^3 = 3xyz \]

### Step 3: Substitute back the values
Substituting back our definitions of \( x, y, z \):
\[ a + b + c = 3 \cdot (a^{1/3} \cdot b^{1/3} \cdot c^{1/3}) \]

### Step 4: Simplify the expression
We can express the product of the cube roots as:
\[ a^{1/3} \cdot b^{1/3} \cdot c^{1/3} = (abc)^{1/3} \]

Thus, we have:
\[ a + b + c = 3(abc)^{1/3} \]

### Step 5: Final conclusion
The relationship we derived shows that:
\[ a + b + c = 3(abc)^{1/3} \]

This implies that the sum of \( a, b, c \) is equal to three times the cube root of the product of \( a, b, c \).

### Conclusion
The correct option based on the derived relationship is:
**Option 4: \( a + b + c = 3(abc)^{1/3} \)**

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