Answer
$$(3v+u)^4=u^4+12u^3v+54u^2v^2+108uv^3+81v^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3v+u)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}u^4+12u^3v+54u^2v^2+108uv^3+81v^4\end{aligned} $$ | |
| ① | $$ (3v+u)^4 = (3v+u)^2 \cdot (3v+u)^2 $$ |
| ② | Find $ \left(3v+u\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3v } $ and $ B = \color{red}{ u }$. $$ \begin{aligned}\left(3v+u\right)^2 = \color{blue}{\left( 3v \right)^2} +2 \cdot 3v \cdot u + \color{red}{u^2} = 9v^2+6uv+u^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{9v^2+6uv+u^2}\right) $ by each term in $ \left( 9v^2+6uv+u^2\right) $. $$ \left( \color{blue}{9v^2+6uv+u^2}\right) \cdot \left( 9v^2+6uv+u^2\right) = \\ = 81v^4+54uv^3+9u^2v^2+54uv^3+36u^2v^2+6u^3v+9u^2v^2+6u^3v+u^4 $$ |
| ④ | Combine like terms: $$ 81v^4+ \color{blue}{54uv^3} + \color{red}{9u^2v^2} + \color{blue}{54uv^3} + \color{green}{36u^2v^2} + \color{orange}{6u^3v} + \color{green}{9u^2v^2} + \color{orange}{6u^3v} +u^4 = \\ = u^4+ \color{orange}{12u^3v} + \color{green}{54u^2v^2} + \color{blue}{108uv^3} +81v^4 $$ |
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