Question
Answer
$$(3+i)^4=96i+28$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3+i)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}i^4+12i^3+54i^2+108i+81 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}1-12i-54+108i+81 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}96i+28\end{aligned} $$ | |
| ① | $$ (3+i)^4 = (3+i)^2 \cdot (3+i)^2 $$ |
| ② | Find $ \left(3+i\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3 } $ and $ B = \color{red}{ i }$. $$ \begin{aligned}\left(3+i\right)^2 = \color{blue}{3^2} +2 \cdot 3 \cdot i + \color{red}{i^2} = 9+6i+i^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{9+6i+i^2}\right) $ by each term in $ \left( 9+6i+i^2\right) $. $$ \left( \color{blue}{9+6i+i^2}\right) \cdot \left( 9+6i+i^2\right) = 81+54i+9i^2+54i+36i^2+6i^3+9i^2+6i^3+i^4 $$ |
| ④ | Combine like terms: $$ 81+ \color{blue}{54i} + \color{red}{9i^2} + \color{blue}{54i} + \color{green}{36i^2} + \color{orange}{6i^3} + \color{green}{9i^2} + \color{orange}{6i^3} +i^4 = \\ = i^4+ \color{orange}{12i^3} + \color{green}{54i^2} + \color{blue}{108i} +81 $$ |
| ⑤ | $$ i^4 = i^2 \cdot i^2 =
( - 1) \cdot ( - 1) =
1 $$ |
| ⑥ | $$ 12i^3 = 12 \cdot \color{blue}{i^2} \cdot i =
12 \cdot ( \color{blue}{-1}) \cdot i =
-12 \cdot \, i $$ |
| ⑦ | $$ 54i^2 = 54 \cdot (-1) = -54 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{-12i} + \color{blue}{108i} \color{red}{-54} + \color{green}{1} + \color{green}{81} = \color{blue}{96i} + \color{green}{28} $$ |
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