Key Concepts and Formulas

• Binomial Expansion: $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
• Imaginary Unit Properties: $i^2 = -1$, $i^3 = -i$
• Complex Number Equality: If $a+bi = c+di$, where $a, b, c, d$ are real numbers, then $a=c$ and $b=d$.

Step-by-Step Solution

1. Simplify the Complex Number We are given the complex number ${\left( { - 2 - {1 \over 3}i} \right)^3}$. We want to simplify the expression inside the parentheses by finding a common denominator. $-2 - \frac{1}{3}i = -\frac{6}{3} - \frac{1}{3}i = -\frac{6+i}{3}$ This simplifies the base of the expression, making the cubing process easier.

2. Cube the Simplified Complex Number Now, we cube the simplified expression: ${\left( { - 2 - {1 \over 3}i} \right)^3} = {\left( {-\frac{6+i}{3}} \right)^3} = -\frac{(6+i)^3}{3^3} = -\frac{(6+i)^3}{27}$ The negative sign comes out because $(-a)^3 = -a^3$.

3. Expand $(6+i)^3$ using the Binomial Theorem We use the binomial expansion formula $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ with $a=6$ and $b=i$: $(6+i)^3 = 6^3 + 3(6^2)(i) + 3(6)(i^2) + i^3$ This expands the complex number raised to the third power.

4. Evaluate Powers of $i$ and Simplify We substitute $i^2 = -1$ and $i^3 = -i$ into the expanded expression: $(6+i)^3 = 216 + 3(36)i + 3(6)(-1) + (-i) = 216 + 108i - 18 - i$ Now, we group the real and imaginary terms: $(6+i)^3 = (216 - 18) + (108 - 1)i = 198 + 107i$ Consolidating the terms into the standard form $A+Bi$ is crucial for the next step.

5. Substitute Back and Compare with the Given Equation We found that ${\left( { - 2 - {1 \over 3}i} \right)^3} = - \frac{{198 + 107i}}{{27}}$. Substituting back into the original equation: ${\left( { - 2 - {1 \over 3}i} \right)^3} = -\frac{198 + 107i}{27} = \frac{-198 - 107i}{27}$ We are given that ${\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}$. Comparing the two expressions, we can equate the numerators: $x + iy = -198 - 107i$ Since $x$ and $y$ are real numbers, we can identify them: $x = -198$ $y = -107$ By expressing the left-hand side in the same format as the right-hand side, we can directly equate the real and imaginary components.

6. Calculate $y - x$ Finally, we substitute the values of $x$ and $y$ into the expression $y - x$: $y - x = -107 - (-198) = -107 + 198 = 91$ This gives us the final answer.

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs, especially when cubing negative numbers and distributing negative signs.
• Powers of $i$: Remember the cycle of powers of $i$: $i, -1, -i, 1$.
• Binomial Expansion: Double-check the coefficients and powers in the binomial expansion.

Summary We simplified the complex number, applied the binomial expansion, and equated the real and imaginary parts to find $x$ and $y$. Finally, we calculated $y-x$ to get $91$.

The final answer is \boxed{91}, which corresponds to option (D).