Question
Answer
$$(4+5i)^3=115i-236$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(4+5i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}64+240i+300i^2+125i^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}64+240i-300-125i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}115i-236\end{aligned} $$ | |
| ① | Find $ \left(4+5i\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 4 $ and $ B = 5i $. $$ \left(4+5i\right)^3 = 4^3+3 \cdot 4^2 \cdot 5i + 3 \cdot 4 \cdot \left( 5i \right)^2+\left( 5i \right)^3 = 64+240i+300i^2+125i^3 $$ |
| ② | $$ 300i^2 = 300 \cdot (-1) = -300 $$ |
| ③ | $$ 125i^3 = 125 \cdot \color{blue}{i^2} \cdot i =
125 \cdot ( \color{blue}{-1}) \cdot i =
-125 \cdot \, i $$ |
| ④ | Combine like terms: $$ \color{blue}{240i} \color{blue}{-125i} \color{red}{-300} + \color{red}{64} = \color{blue}{115i} \color{red}{-236} $$ |
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