Answer
$$(2i+2)^4=-64$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2i+2)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}16i^4+64i^3+96i^2+64i+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}16-64i-96+64i+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-64\end{aligned} $$ | |
| ① | $$ (2i+2)^4 = (2i+2)^2 \cdot (2i+2)^2 $$ |
| ② | Find $ \left(2i+2\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 2i } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(2i+2\right)^2 = \color{blue}{\left( 2i \right)^2} +2 \cdot 2i \cdot 2 + \color{red}{2^2} = 4i^2+8i+4\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{4i^2+8i+4}\right) $ by each term in $ \left( 4i^2+8i+4\right) $. $$ \left( \color{blue}{4i^2+8i+4}\right) \cdot \left( 4i^2+8i+4\right) = 16i^4+32i^3+16i^2+32i^3+64i^2+32i+16i^2+32i+16 $$ |
| ④ | Combine like terms: $$ 16i^4+ \color{blue}{32i^3} + \color{red}{16i^2} + \color{blue}{32i^3} + \color{green}{64i^2} + \color{orange}{32i} + \color{green}{16i^2} + \color{orange}{32i} +16 = \\ = 16i^4+ \color{blue}{64i^3} + \color{green}{96i^2} + \color{orange}{64i} +16 $$ |
| ⑤ | $$ 16i^4 = 16 \cdot i^2 \cdot i^2 =
16 \cdot ( - 1) \cdot ( - 1) =
16 $$ |
| ⑥ | $$ 64i^3 = 64 \cdot \color{blue}{i^2} \cdot i =
64 \cdot ( \color{blue}{-1}) \cdot i =
-64 \cdot \, i $$ |
| ⑦ | $$ 96i^2 = 96 \cdot (-1) = -96 $$ |
| ⑧ | Combine like terms: $$ \, \color{blue}{ -\cancel{64i}} \,+ \, \color{blue}{ \cancel{64i}} \, \color{green}{-96} + \color{orange}{16} + \color{orange}{16} = \color{orange}{-64} $$ |
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