Question
Answer
$$(1+4i)^4(1-4i)^4=-57600i^2+25921$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+4i)^4(1-4i)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(256i^4+256i^3+96i^2+16i+1)(256i^4-256i^3+96i^2-16i+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}(256-256i-96+16i+1)(256+256i-96-16i+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} } }}}(-240i+161)(240i+161) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle14}{\textcircled {14}} } }}}-57600i^2-38640i+38640i+25921 \xlongequal{ } \\[1 em] & \xlongequal{ }-57600i^2 -\cancel{38640i}+ \cancel{38640i}+25921 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle15}{\textcircled {15}} } }}}-57600i^2+25921\end{aligned} $$ | |
| ① | $$ (1+4i)^4 = (1+4i)^2 \cdot (1+4i)^2 $$ |
| ② | Find $ \left(1+4i\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ 4i }$. $$ \begin{aligned}\left(1+4i\right)^2 = \color{blue}{1^2} +2 \cdot 1 \cdot 4i + \color{red}{\left( 4i \right)^2} = 1+8i+16i^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{1+8i+16i^2}\right) $ by each term in $ \left( 1+8i+16i^2\right) $. $$ \left( \color{blue}{1+8i+16i^2}\right) \cdot \left( 1+8i+16i^2\right) = 1+8i+16i^2+8i+64i^2+128i^3+16i^2+128i^3+256i^4 $$ |
| ④ | Combine like terms: $$ 1+ \color{blue}{8i} + \color{red}{16i^2} + \color{blue}{8i} + \color{green}{64i^2} + \color{orange}{128i^3} + \color{green}{16i^2} + \color{orange}{128i^3} +256i^4 = \\ = 256i^4+ \color{orange}{256i^3} + \color{green}{96i^2} + \color{blue}{16i} +1 $$$$ (1-4i)^4 = (1-4i)^2 \cdot (1-4i)^2 $$ |
| ⑤ | Find $ \left(1-4i\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ 4i }$. $$ \begin{aligned}\left(1-4i\right)^2 = \color{blue}{1^2} -2 \cdot 1 \cdot 4i + \color{red}{\left( 4i \right)^2} = 1-8i+16i^2\end{aligned} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{1-8i+16i^2}\right) $ by each term in $ \left( 1-8i+16i^2\right) $. $$ \left( \color{blue}{1-8i+16i^2}\right) \cdot \left( 1-8i+16i^2\right) = 1-8i+16i^2-8i+64i^2-128i^3+16i^2-128i^3+256i^4 $$ |
| ⑦ | Combine like terms: $$ 1 \color{blue}{-8i} + \color{red}{16i^2} \color{blue}{-8i} + \color{green}{64i^2} \color{orange}{-128i^3} + \color{green}{16i^2} \color{orange}{-128i^3} +256i^4 = \\ = 256i^4 \color{orange}{-256i^3} + \color{green}{96i^2} \color{blue}{-16i} +1 $$ |
| ⑧ | $$ 256i^4 = 256 \cdot i^2 \cdot i^2 =
256 \cdot ( - 1) \cdot ( - 1) =
256 $$ |
| ⑨ | $$ 256i^3 = 256 \cdot \color{blue}{i^2} \cdot i =
256 \cdot ( \color{blue}{-1}) \cdot i =
-256 \cdot \, i $$ |
| ⑩ | $$ 96i^2 = 96 \cdot (-1) = -96 $$$$ 256i^4 = 256 \cdot i^2 \cdot i^2 =
256 \cdot ( - 1) \cdot ( - 1) =
256 $$ |
| ⑪ | $$ -256i^3 = -256 \cdot \color{blue}{i^2} \cdot i =
-256 \cdot ( \color{blue}{-1}) \cdot i =
256 \cdot \, i $$ |
| ⑫ | $$ 96i^2 = 96 \cdot (-1) = -96 $$ |
| ⑬ | Combine like terms: $$ \color{blue}{256} \color{red}{-256i} \color{green}{-96} + \color{red}{16i} + \color{green}{1} = \color{red}{-240i} + \color{green}{161} $$Combine like terms: $$ \color{blue}{256} + \color{red}{256i} \color{green}{-96} \color{red}{-16i} + \color{green}{1} = \color{red}{240i} + \color{green}{161} $$ |
| ⑭ | Multiply each term of $ \left( \color{blue}{-240i+161}\right) $ by each term in $ \left( 240i+161\right) $. $$ \left( \color{blue}{-240i+161}\right) \cdot \left( 240i+161\right) = -57600i^2 -\cancel{38640i}+ \cancel{38640i}+25921 $$ |
| ⑮ | Combine like terms: $$ -57600i^2 \, \color{blue}{ -\cancel{38640i}} \,+ \, \color{blue}{ \cancel{38640i}} \,+25921 = -57600i^2+25921 $$ |
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