If `a^(3) + b^(3)=0` find `a+b=`

Options

A. `sqrt(3ab)`
B. `sqrt(2ab)`
C. 3ab
D. `sqrt(4ab)`

Answer

To solve the equation \( a^3 + b^3 = 0 \) and find \( a + b \), we can follow these steps:

### Step 1: Use the identity for the sum of cubes
We know that the sum of cubes can be factored using the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Given that \( a^3 + b^3 = 0 \), we can set up the equation:
\[ (a + b)(a^2 - ab + b^2) = 0 \]

### Step 2: Analyze the factors
For the product to be zero, at least one of the factors must be zero. This gives us two cases to consider:
1. \( a + b = 0 \)
2. \( a^2 - ab + b^2 = 0 \)

### Step 3: Solve the first case
If \( a + b = 0 \), then we can conclude that:
\[ a + b = 0 \]

### Step 4: Solve the second case
Now, let's analyze the second factor \( a^2 - ab + b^2 = 0 \). This is a quadratic equation in terms of \( a \) and \( b \). However, we can see that for real numbers \( a \) and \( b \), this equation does not yield any real solutions because the discriminant is negative.

### Conclusion
Thus, the only solution we have is from the first case:
\[ a + b = 0 \]

### Final Answer
Therefore, the value of \( a + b \) is:
\[ \boxed{0} \]

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