Question
Answer
$$(a+bi)^4=b^4i^4+4ab^3i^3+6a^2b^2i^2+4a^3bi+a^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a+bi)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}b^4i^4+4ab^3i^3+6a^2b^2i^2+4a^3bi+a^4\end{aligned} $$ | |
| ① | $$ (a+bi)^4 = (a+bi)^2 \cdot (a+bi)^2 $$ |
| ② | Find $ \left(a+bi\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ a } $ and $ B = \color{red}{ bi }$. $$ \begin{aligned}\left(a+bi\right)^2 = \color{blue}{a^2} +2 \cdot a \cdot bi + \color{red}{\left( bi \right)^2} = a^2+2abi+b^2i^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{a^2+2abi+b^2i^2}\right) $ by each term in $ \left( a^2+2abi+b^2i^2\right) $. $$ \left( \color{blue}{a^2+2abi+b^2i^2}\right) \cdot \left( a^2+2abi+b^2i^2\right) = \\ = a^4+2a^3bi+a^2b^2i^2+2a^3bi+4a^2b^2i^2+2ab^3i^3+a^2b^2i^2+2ab^3i^3+b^4i^4 $$ |
| ④ | Combine like terms: $$ a^4+ \color{blue}{2a^3bi} + \color{red}{a^2b^2i^2} + \color{blue}{2a^3bi} + \color{green}{4a^2b^2i^2} + \color{orange}{2ab^3i^3} + \color{green}{a^2b^2i^2} + \color{orange}{2ab^3i^3} +b^4i^4 = \\ = b^4i^4+ \color{orange}{4ab^3i^3} + \color{green}{6a^2b^2i^2} + \color{blue}{4a^3bi} +a^4 $$ |
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