The roots of the cubic equation `(z+ab)^3=a^3,a !=0` represents the vertices of an equilateral triangle of sides of length

Options

A. `(1)/(sqrt(3))|ab|`
B. `sqrt(3)|a|`
C. `sqrt(3)|b|`
D. `|a|`

Explanation

To solve the problem, we need to analyze the cubic equation given and find the roots, which represent the vertices of an equilateral triangle. Let's go through the solution step by step.

### Step 1: Rewrite the given equation
We start with the cubic equation:
\[ (z + ab)^3 = a^3 \]
To find the roots, we can take the cube root of both sides.

### Step 2: Take the cube root
Taking the cube root gives us:
\[ z + ab = a \omega^k \quad \text{for } k = 0, 1, 2 \]
where \(\omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\) is a primitive cube root of unity.

### Step 3: Solve for \(z\)
Now we can express \(z\) in terms of \(\omega\):
1. For \(k = 0\):
\[ z_1 = a - ab \]
2. For \(k = 1\):
\[ z_2 = a \omega - ab = a \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) - ab \]
Simplifying gives:
\[ z_2 = -\frac{a}{2} + i\frac{a\sqrt{3}}{2} - ab \]
3. For \(k = 2\):
\[ z_3 = a \omega^2 - ab = a \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) - ab \]
Simplifying gives:
\[ z_3 = -\frac{a}{2} - i\frac{a\sqrt{3}}{2} - ab \]

### Step 4: Calculate the sides of the triangle
Now, we need to find the lengths of the sides of the triangle formed by these points:
- The distance between \(z_1\) and \(z_2\):
\[ z_1 - z_2 = \left(a - ab\right) - \left(-\frac{a}{2} + i\frac{a\sqrt{3}}{2} - ab\right) \]
This simplifies to:
\[ z_1 - z_2 = a + \frac{a}{2} - i\frac{a\sqrt{3}}{2} = \frac{3a}{2} - i\frac{a\sqrt{3}}{2} \]
The modulus (length of the side) is:
\[ |z_1 - z_2| = \sqrt{\left(\frac{3a}{2}\right)^2 + \left(-\frac{a\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9a^2}{4} + \frac{3a^2}{4}} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 5: Conclusion
Since the triangle is equilateral, all sides are equal. Therefore, the length of each side of the triangle is:
\[ \text{Length of each side} = a\sqrt{3} \]

### Final Answer
The roots of the cubic equation represent the vertices of an equilateral triangle of sides of length:
\[ \sqrt{3} \cdot |a| \]