Question
Answer
$$(-3+2i)^3=46i+9$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-3+2i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}8i^3-36i^2+54i-27 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-8i+36+54i-27 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}46i+9\end{aligned} $$ | |
| ① | Find $ \left(-3+2i\right)^3 $ in two steps. S1: Swap two terms inside bracket S2: apply formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 2i $ and $ B = 3 $. $$ \left(-3+2i\right)^3 \xlongequal{ S1 } \left(2i-3\right)^3 = \left( 2i \right)^3-3 \cdot \left( 2i \right)^2 \cdot 3 + 3 \cdot 2i \cdot 3^2-3^3 = 8i^3-36i^2+54i-27 $$ |
| ② | $$ 8i^3 = 8 \cdot \color{blue}{i^2} \cdot i =
8 \cdot ( \color{blue}{-1}) \cdot i =
-8 \cdot \, i $$ |
| ③ | $$ -36i^2 = -36 \cdot (-1) = 36 $$ |
| ④ | Combine like terms: $$ \color{blue}{-8i} + \color{blue}{54i} + \color{red}{36} \color{red}{-27} = \color{blue}{46i} + \color{red}{9} $$ |
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