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