Let `(-2 -1/3 i)^3=(x+iy)/27 (i= sqrt(-1))` where x and y are real numbers then y-x equals

Options

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

Answer

To solve the equation \((-2 - \frac{1}{3} i)^3 = \frac{x + iy}{27}\), where \(i = \sqrt{-1}\), we will follow these steps:

### Step 1: Calculate \((-2 - \frac{1}{3} i)^3\)

Using the binomial expansion formula, we can express \((a + b)^3\) as:
\[ a^3 + 3a^2b + 3ab^2 + b^3 \]
Here, let \(a = -2\) and \(b = -\frac{1}{3} i\).

Calculating each term:
1. \(a^3 = (-2)^3 = -8\)
2. \(b^3 = \left(-\frac{1}{3} i\right)^3 = -\frac{1}{27} i^3 = -\frac{1}{27}(-i) = \frac{i}{27}\)
3. \(3a^2b = 3(-2)^2\left(-\frac{1}{3} i\right) = 3 \cdot 4 \cdot \left(-\frac{1}{3} i\right) = -4i\)
4. \(3ab^2 = 3(-2)\left(-\frac{1}{3} i\right)^2 = 3(-2)\left(\frac{1}{9}\right)(-1) = \frac{6}{9} = \frac{2}{3}\)

Now, combining all these:
\[ (-2 - \frac{1}{3} i)^3 = -8 + \frac{2}{3} - 4i + \frac{i}{27} \]

### Step 2: Combine the Real and Imaginary Parts

To combine the real parts:
\[ -8 + \frac{2}{3} = -\frac{24}{3} + \frac{2}{3} = -\frac{22}{3} \]

For the imaginary parts:
\[ -4i + \frac{i}{27} = -\frac{108}{27}i + \frac{1}{27}i = -\frac{107}{27}i \]

So, we have:
\[ (-2 - \frac{1}{3} i)^3 = -\frac{22}{3} - \frac{107}{27}i \]

### Step 3: Set the Equation Equal to \(\frac{x + iy}{27}\)

We can write:
\[ -\frac{22}{3} - \frac{107}{27}i = \frac{x + iy}{27} \]

Multiplying both sides by 27 gives:
\[ -22 - \frac{107}{27} \cdot 27i = x + iy \]

This simplifies to:
\[ -22 - 107i = x + iy \]

### Step 4: Equate Real and Imaginary Parts

From here, we can equate the real and imaginary parts:
1. Real part: \(x = -22\)
2. Imaginary part: \(y = -107\)

### Step 5: Calculate \(y - x\)

Now, we need to find \(y - x\):
\[ y - x = -107 - (-22) = -107 + 22 = -85 \]

### Final Answer

Thus, the value of \(y - x\) is:
\[ \boxed{-85} \]