Question
Answer
$$(m^2-n^2)^4+4m^2n^2(m^2-n^2)^2+8m^4n^4=m^8+6m^4n^4+n^8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(m^2-n^2)^4+4m^2n^2(m^2-n^2)^2+8m^4n^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}m^8-4m^6n^2+6m^4n^4-4m^2n^6+n^8+4m^2n^2(1m^4-2m^2n^2+n^4)+8m^4n^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}m^8-4m^6n^2+6m^4n^4-4m^2n^6+n^8+4m^6n^2-8m^4n^4+4m^2n^6+8m^4n^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}m^8-2m^4n^4+n^8+8m^4n^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}m^8+6m^4n^4+n^8\end{aligned} $$ | |
| ① | $$ (m^2-n^2)^4 = (m^2-n^2)^2 \cdot (m^2-n^2)^2 $$ |
| ② | Find $ \left(m^2-n^2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ m^2 } $ and $ B = \color{red}{ n^2 }$. $$ \begin{aligned}\left(m^2-n^2\right)^2 = \color{blue}{\left( m^2 \right)^2} -2 \cdot m^2 \cdot n^2 + \color{red}{\left( n^2 \right)^2} = m^4-2m^2n^2+n^4\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{m^4-2m^2n^2+n^4}\right) $ by each term in $ \left( m^4-2m^2n^2+n^4\right) $. $$ \left( \color{blue}{m^4-2m^2n^2+n^4}\right) \cdot \left( m^4-2m^2n^2+n^4\right) = \\ = m^8-2m^6n^2+m^4n^4-2m^6n^2+4m^4n^4-2m^2n^6+m^4n^4-2m^2n^6+n^8 $$ |
| ④ | Combine like terms: $$ m^8 \color{blue}{-2m^6n^2} + \color{red}{m^4n^4} \color{blue}{-2m^6n^2} + \color{green}{4m^4n^4} \color{orange}{-2m^2n^6} + \color{green}{m^4n^4} \color{orange}{-2m^2n^6} +n^8 = \\ = m^8 \color{blue}{-4m^6n^2} + \color{green}{6m^4n^4} \color{orange}{-4m^2n^6} +n^8 $$Find $ \left(m^2-n^2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ m^2 } $ and $ B = \color{red}{ n^2 }$. $$ \begin{aligned}\left(m^2-n^2\right)^2 = \color{blue}{\left( m^2 \right)^2} -2 \cdot m^2 \cdot n^2 + \color{red}{\left( n^2 \right)^2} = m^4-2m^2n^2+n^4\end{aligned} $$ |
| ⑤ | Multiply $ \color{blue}{4m^2n^2} $ by $ \left( m^4-2m^2n^2+n^4\right) $ $$ \color{blue}{4m^2n^2} \cdot \left( m^4-2m^2n^2+n^4\right) = 4m^6n^2-8m^4n^4+4m^2n^6 $$ |
| ⑥ | Combine like terms: $$ m^8 \, \color{blue}{ -\cancel{4m^6n^2}} \,+ \color{green}{6m^4n^4} \, \color{orange}{ -\cancel{4m^2n^6}} \,+n^8+ \, \color{blue}{ \cancel{4m^6n^2}} \, \color{green}{-8m^4n^4} + \, \color{orange}{ \cancel{4m^2n^6}} \, = m^8 \color{green}{-2m^4n^4} +n^8 $$ |
| ⑦ | Combine like terms: $$ m^8 \color{blue}{-2m^4n^4} +n^8+ \color{blue}{8m^4n^4} = m^8+ \color{blue}{6m^4n^4} +n^8 $$ |
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