Discussion

If \(x, y \in \mathbb{R}\) and \(x + iy = -(6 + i)^{3}, i^{2} = -1\), then \(x - y\) is equal to

Answer: D

\(x - y\) equals −91.

Expand by the binomial theorem:

\[ \begin{aligned} (6 + i)^{3} &= 216 + 3(36)i + 3(6)i^{2} + i^{3} \\ &= 216 + 108i - 18 - i = 198 + 107i \end{aligned} \]

So \(x + iy = -198 - 107i\), giving \(x = -198\), \(y = -107\) and \(x - y = -198 + 107 = -91\).

91 drops the overall minus sign, −107 is just y, and ±93 comes from a slip in the middle terms.

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