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

Explanation

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: Apply the Binomial Theorem
We can expand \((-2 - \frac{1}{3} i)^3\) using the binomial theorem:
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
Here, \(a = -2\) and \(b = -\frac{1}{3} i\).

### Step 2: Calculate \(a^3\)
Calculate \((-2)^3\):
\[ (-2)^3 = -8 \]

### Step 3: Calculate \(b^3\)
Calculate \(\left(-\frac{1}{3} i\right)^3\):
\[ \left(-\frac{1}{3} i\right)^3 = -\frac{1}{27} i^3 = -\frac{1}{27} (-i) = \frac{1}{27} i \]

### Step 4: Calculate \(3a^2b\)
Calculate \(3(-2)^2\left(-\frac{1}{3} i\right)\):
\[ 3(-2)^2\left(-\frac{1}{3} i\right) = 3 \cdot 4 \cdot \left(-\frac{1}{3} i\right) = -4i \]

### Step 5: Calculate \(3ab^2\)
Calculate \(3(-2)\left(-\frac{1}{3} i\right)^2\):
\[ 3(-2)\left(-\frac{1}{3} i\right)^2 = 3(-2)\left(\frac{1}{9}(-1)\right) = \frac{6}{9} = \frac{2}{3} \]

### Step 6: Combine all parts
Now, combine all the parts:
\[ (-2 - \frac{1}{3} i)^3 = -8 + \frac{1}{27} i - 4i + \frac{2}{3} \]
Combine the real parts and the imaginary parts:
\[ = \left(-8 + \frac{2}{3}\right) + \left(\frac{1}{27} - 4\right)i \]

### Step 7: Simplify the real part
Convert \(-8\) to a fraction:
\[ -8 = -\frac{24}{3} \quad \Rightarrow \quad -\frac{24}{3} + \frac{2}{3} = -\frac{22}{3} \]

### Step 8: Simplify the imaginary part
Convert \(4\) to a fraction:
\[ 4 = \frac{108}{27} \quad \Rightarrow \quad \frac{1}{27} - \frac{108}{27} = -\frac{107}{27} \]

### Step 9: Write the final expression
Thus, we have:
\[ (-2 - \frac{1}{3} i)^3 = \left(-\frac{22}{3} - \frac{107}{27} i\right) \]

### Step 10: Set equal to \(\frac{x + iy}{27}\)
From the equation \((-2 - \frac{1}{3} i)^3 = \frac{x + iy}{27}\), we can multiply both sides by 27:
\[ -22 + \frac{22}{3} \cdot 27 = x + iy \]
Thus, we have:
\[ x = -198, \quad y = -107 \]

### Step 11: Calculate \(y - x\)
Now, calculate \(y - x\):
\[ y - x = -107 - (-198) = -107 + 198 = 91 \]

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

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