Answer
$$(-1-6i)^3=198i+107$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-1-6i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}-1-18i-108i^2-216i^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-1-18i+108+216i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}198i+107\end{aligned} $$ | |
| ① | Find $ \left(-1-6i\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = -1 $ and $ B = 6i $. $$ \left(-1-6i\right)^3 = \left( -1 \right)^3-3 \cdot \left( -1 \right)^2 \cdot 6i + 3 \cdot \left( -1 \right) \cdot \left( 6i \right)^2-\left( 6i \right)^3 = -1-18i-108i^2-216i^3 $$ |
| ② | $$ -108i^2 = -108 \cdot (-1) = 108 $$ |
| ③ | $$ -216i^3 = -216 \cdot \color{blue}{i^2} \cdot i =
-216 \cdot ( \color{blue}{-1}) \cdot i =
216 \cdot \, i $$ |
| ④ | Combine like terms: $$ \color{blue}{-18i} + \color{blue}{216i} + \color{red}{108} \color{red}{-1} = \color{blue}{198i} + \color{red}{107} $$ |
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