If `a^((1)/(3)) + b^((1)/(3)) + c^((1)/(3)) = 0`, then `(a+b+c)^(6)` is equal to :

Options

A. 81 abc
B. `729 a^(2)b^(2)c^(2)`
C. 729 abc
D. `81a^(2)b^(2)c^(2)`

Answer

To solve the problem, we start with the given equation:

\[ a^{\frac{1}{3}} + b^{\frac{1}{3}} + c^{\frac{1}{3}} = 0 \]

### Step 1: Recognize the identity
From algebra, we know that if \( x + y + z = 0 \), then:

\[ x^3 + y^3 + z^3 = 3xyz \]

In our case, we can let \( x = a^{\frac{1}{3}}, y = b^{\frac{1}{3}}, z = c^{\frac{1}{3}} \). Thus, we can write:

\[ a + b + c = (a^{\frac{1}{3}})^3 + (b^{\frac{1}{3}})^3 + (c^{\frac{1}{3}})^3 = 3(a^{\frac{1}{3}})(b^{\frac{1}{3}})(c^{\frac{1}{3}}) \]

### Step 2: Simplify the expression
This means we can express \( a + b + c \) in terms of \( abc \):

\[ a + b + c = 3 \cdot (a^{\frac{1}{3}} b^{\frac{1}{3}} c^{\frac{1}{3}}) \]

### Step 3: Calculate \( (a + b + c)^6 \)
Now we need to find \( (a + b + c)^6 \):

\[ (a + b + c)^6 = (3(a^{\frac{1}{3}} b^{\frac{1}{3}} c^{\frac{1}{3}}))^6 \]

### Step 4: Expand the expression
Using the property of exponents, we can simplify this further:

\[ = 3^6 \cdot (a^{\frac{1}{3}} b^{\frac{1}{3}} c^{\frac{1}{3}})^6 \]

### Step 5: Simplify the powers
Now, simplifying \( (a^{\frac{1}{3}} b^{\frac{1}{3}} c^{\frac{1}{3}})^6 \):

\[ = 3^6 \cdot a^2 b^2 c^2 \]

### Step 6: Calculate \( 3^6 \)
Calculating \( 3^6 \):

\[ 3^6 = 729 \]

### Final Result
Thus, we have:

\[ (a + b + c)^6 = 729 \cdot abc \]

So, the final answer is:

\[ \boxed{729abc} \]