Question
Answer
$$(4+3i)^2-i^2(i^6-2)=9i^2+24i+13$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(4+3i)^2-i^2(i^6-2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(4+3i)^2--1\cdot(-1-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(4+3i)^2--1\cdot(-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(4+3i)^2-3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}16+24i+9i^2-3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}9i^2+24i+13\end{aligned} $$ | |
| ① | $$ i^2 = -1 $$$$ i^6 = i^{4 \cdot 1 + 2} =
\left( i^4 \right)^{ 1 } \cdot i^2 =
1^{ 1 } \cdot (-1) =
-1 = -1 $$ |
| ② | Combine like terms: $$ \color{blue}{-1} \color{blue}{-2} = \color{blue}{-3} $$ |
| ③ | $$ -1 \cdot -3 = 3 $$ |
| ④ | Find $ \left(4+3i\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 4 } $ and $ B = \color{red}{ 3i }$. $$ \begin{aligned}\left(4+3i\right)^2 = \color{blue}{4^2} +2 \cdot 4 \cdot 3i + \color{red}{\left( 3i \right)^2} = 16+24i+9i^2\end{aligned} $$ |
| ⑤ | Combine like terms: $$ \color{blue}{16} +24i+9i^2 \color{blue}{-3} = 9i^2+24i+ \color{blue}{13} $$ |
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