Let $$\left(-2 - \frac{1}{3}i\right)^3 = \frac{x+iy}{27}$$ $$(i = \sqrt{-1})$$, where $$x$$ and $$y$$ are real numbers then $$y - x$$ equals

We begin with the complex number $$\left(-2-\dfrac13\,i\right)$$ and we have to raise it to the third power. A very convenient first observation is that a common denominator of 3 can be taken out:

$$\left(-2-\dfrac13\,i\right)=\dfrac{-6-i}{3}.$$

Raising both sides to the third power gives

$$\left(-2-\dfrac13\,i\right)^3=\left(\dfrac{-6-i}{3}\right)^3=\dfrac{(-6-i)^3}{27}.$$

The question itself states that

$$\left(-2-\dfrac13\,i\right)^3=\dfrac{x+iy}{27},$$

so it is clear that we must evaluate $$(-6-i)^3$$; whatever real part and imaginary part we get will be the required $$x$$ and $$y$$ respectively.

To cube $$(-6-i)$$ we can first square it and then multiply once more by $$(-6-i)$$. Throughout, we keep the fundamental multiplication rule for complex numbers in mind:

$$ (a+ib)(c+id)=(ac-bd)+i(ad+bc). $$

Step 1: Calculate $$(-6-i)^2$$.

Using the ordinary algebraic expansion $$(A+B)^2=A^2+2AB+B^2$$ with $$A=-6$$ and $$B=-i$$,

$$(-6-i)^2=(-6)^2+2(-6)(-i)+(-i)^2.$$

Now compute each term carefully:

$$(-6)^2=36,$$

$$2(-6)(-i)=12i,$$

$$(-i)^2=i^2=(-1)= -1.$$

Adding them up,

$$(-6-i)^2=36+12i-1=35+12i.$$

Step 2: Multiply this square by $$(-6-i)$$ to obtain the cube.

We now multiply $$(35+12i)$$ by $$(-6-i)$$, applying the multiplication formula stated earlier. Identify

$$c=35,\quad d=12,\quad a=-6,\quad b=-1.$$

Then

$$\bigl(35+12i\bigr)\bigl(-6-i\bigr)=(ca-db)+i(cb+da).$$

Compute the real part first:

$$ca=35\times(-6)=-210,$$

$$db=12\times(-1)=-12,$$

$$ca-db=-210-(-12)=-210+12=-198.$$

Next compute the imaginary coefficient:

$$cb=35\times(-1)=-35,$$

$$da=12\times(-6)=-72,$$

$$cb+da=-35+(-72)=-107.$$

Thus

$$(-6-i)^3=-198-107\,i.$$

Step 3: Relate this to the required form.

Recall that

$$\left(-2-\dfrac13\,i\right)^3=\dfrac{(-6-i)^3}{27},$$

so substituting our result gives

$$\left(-2-\dfrac13\,i\right)^3=\dfrac{-198-107\,i}{27}.$$

Comparing this with the expression $$\dfrac{x+iy}{27}$$ we at once read off

$$x=-198,\qquad y=-107.$$

Step 4: Compute $$y-x$$.

$$y-x=(-107)-(-198)=-107+198=91.$$

Hence, the correct answer is Option A.