Answer
$$(u-4v)^4=u^4-16u^3v+96u^2v^2-256uv^3+256v^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(u-4v)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}u^4-16u^3v+96u^2v^2-256uv^3+256v^4\end{aligned} $$ | |
| ① | $$ (u-4v)^4 = (u-4v)^2 \cdot (u-4v)^2 $$ |
| ② | Find $ \left(u-4v\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ u } $ and $ B = \color{red}{ 4v }$. $$ \begin{aligned}\left(u-4v\right)^2 = \color{blue}{u^2} -2 \cdot u \cdot 4v + \color{red}{\left( 4v \right)^2} = u^2-8uv+16v^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{u^2-8uv+16v^2}\right) $ by each term in $ \left( u^2-8uv+16v^2\right) $. $$ \left( \color{blue}{u^2-8uv+16v^2}\right) \cdot \left( u^2-8uv+16v^2\right) = \\ = u^4-8u^3v+16u^2v^2-8u^3v+64u^2v^2-128uv^3+16u^2v^2-128uv^3+256v^4 $$ |
| ④ | Combine like terms: $$ u^4 \color{blue}{-8u^3v} + \color{red}{16u^2v^2} \color{blue}{-8u^3v} + \color{green}{64u^2v^2} \color{orange}{-128uv^3} + \color{green}{16u^2v^2} \color{orange}{-128uv^3} +256v^4 = \\ = u^4 \color{blue}{-16u^3v} + \color{green}{96u^2v^2} \color{orange}{-256uv^3} +256v^4 $$ |
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