If `(-2-(1)/(3)i)^(3)= (x+iy)/(27),xy, in R` then y-x equals

Options

A. 91
B. 85
C. `-85`
D. `-91`

Answer

To solve the problem, we need to evaluate the expression \((-2 - \frac{1}{3}i)^3\) and express it in the form \(\frac{x + iy}{27}\), where \(x\) and \(y\) are real numbers. We will then find \(y - x\).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ (-2 - \frac{1}{3}i)^3 \]
To simplify calculations, we can express this as:
\[ \left(-\frac{6 + i}{3}\right)^3 = \frac{-(6 + i)^3}{27} \]

2. **Expand Using the Binomial Theorem**:
We need to expand \((6 + i)^3\) using the binomial theorem:
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
Here, \(a = 6\) and \(b = i\):
\[ (6 + i)^3 = 6^3 + 3(6^2)(i) + 3(6)(i^2) + i^3 \]

3. **Calculate Each Term**:
- \(6^3 = 216\)
- \(3(6^2)(i) = 3 \cdot 36 \cdot i = 108i\)
- \(3(6)(i^2) = 3 \cdot 6 \cdot (-1) = -18\)
- \(i^3 = -i\)

Combining these, we have:
\[ (6 + i)^3 = 216 + 108i - 18 - i = 198 + 107i \]

4. **Substitute Back**:
Now substituting back into our expression:
\[ (-2 - \frac{1}{3}i)^3 = \frac{-(198 + 107i)}{27} = \frac{-198 - 107i}{27} \]

5. **Identify \(x\) and \(y\)**:
From the expression \(\frac{x + iy}{27}\), we can identify:
\[ x = -198 \quad \text{and} \quad y = -107 \]

6. **Calculate \(y - x\)**:
Now we find \(y - x\):
\[ y - x = -107 - (-198) = -107 + 198 = 91 \]

### Final Answer:
Thus, \(y - x = 91\).