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