Question
Answer
$$3(u^4-2v^4)(9u^2-8v^2)-2(3u^3-4v^3)^2=9u^6-24u^4v^2+48u^3v^3-54u^2v^4+16v^6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3(u^4-2v^4)(9u^2-8v^2)-2(3u^3-4v^3)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}3(u^4-2v^4)(9u^2-8v^2)-2(9u^6-24u^3v^3+16v^6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(3u^4-6v^4)(9u^2-8v^2)-(18u^6-48u^3v^3+32v^6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}27u^6-24u^4v^2-54u^2v^4+48v^6-(18u^6-48u^3v^3+32v^6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}27u^6-24u^4v^2-54u^2v^4+48v^6-18u^6+48u^3v^3-32v^6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}9u^6-24u^4v^2+48u^3v^3-54u^2v^4+16v^6\end{aligned} $$ | |
| ① | Find $ \left(3u^3-4v^3\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3u^3 } $ and $ B = \color{red}{ 4v^3 }$. $$ \begin{aligned}\left(3u^3-4v^3\right)^2 = \color{blue}{\left( 3u^3 \right)^2} -2 \cdot 3u^3 \cdot 4v^3 + \color{red}{\left( 4v^3 \right)^2} = 9u^6-24u^3v^3+16v^6\end{aligned} $$ |
| ② | Multiply $ \color{blue}{3} $ by $ \left( u^4-2v^4\right) $ $$ \color{blue}{3} \cdot \left( u^4-2v^4\right) = 3u^4-6v^4 $$Multiply $ \color{blue}{2} $ by $ \left( 9u^6-24u^3v^3+16v^6\right) $ $$ \color{blue}{2} \cdot \left( 9u^6-24u^3v^3+16v^6\right) = 18u^6-48u^3v^3+32v^6 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{3u^4-6v^4}\right) $ by each term in $ \left( 9u^2-8v^2\right) $. $$ \left( \color{blue}{3u^4-6v^4}\right) \cdot \left( 9u^2-8v^2\right) = 27u^6-24u^4v^2-54u^2v^4+48v^6 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 18u^6-48u^3v^3+32v^6 \right) = -18u^6+48u^3v^3-32v^6 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{27u^6} -24u^4v^2-54u^2v^4+ \color{red}{48v^6} \color{blue}{-18u^6} +48u^3v^3 \color{red}{-32v^6} = \color{blue}{9u^6} -24u^4v^2+48u^3v^3-54u^2v^4+ \color{red}{16v^6} $$ |
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