Question
Answer
$$(a^4+2b^4)^4-(a^4-2b^4)^4=16a^{12}b^4+64a^4b^{12}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a^4+2b^4)^4-(a^4-2b^4)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}a^{16}+8a^{12}b^4+24a^8b^8+32a^4b^{12}+16b^{16}-(1a^{16}-8a^{12}b^4+24a^8b^8-32a^4b^{12}+16b^{16}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}a^{16}+8a^{12}b^4+24a^8b^8+32a^4b^{12}+16b^{16}-a^{16}+8a^{12}b^4-24a^8b^8+32a^4b^{12}-16b^{16} \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{a^{16}}+8a^{12}b^4+ \cancel{24a^8b^8}+32a^4b^{12}+ \cancel{16b^{16}} -\cancel{a^{16}}+8a^{12}b^4 -\cancel{24a^8b^8}+32a^4b^{12} -\cancel{16b^{16}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}16a^{12}b^4+64a^4b^{12}\end{aligned} $$ | |
| ① | $$ (a^4+2b^4)^4 = (a^4+2b^4)^2 \cdot (a^4+2b^4)^2 $$ |
| ② | Find $ \left(a^4+2b^4\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^4 } $ and $ B = \color{red}{ 2b^4 }$. $$ \begin{aligned}\left(a^4+2b^4\right)^2 = \color{blue}{\left( a^4 \right)^2} +2 \cdot a^4 \cdot 2b^4 + \color{red}{\left( 2b^4 \right)^2} = a^8+4a^4b^4+4b^8\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{a^8+4a^4b^4+4b^8}\right) $ by each term in $ \left( a^8+4a^4b^4+4b^8\right) $. $$ \left( \color{blue}{a^8+4a^4b^4+4b^8}\right) \cdot \left( a^8+4a^4b^4+4b^8\right) = \\ = a^{16}+4a^{12}b^4+4a^8b^8+4a^{12}b^4+16a^8b^8+16a^4b^{12}+4a^8b^8+16a^4b^{12}+16b^{16} $$ |
| ④ | Combine like terms: $$ a^{16}+ \color{blue}{4a^{12}b^4} + \color{red}{4a^8b^8} + \color{blue}{4a^{12}b^4} + \color{green}{16a^8b^8} + \color{orange}{16a^4b^{12}} + \color{green}{4a^8b^8} + \color{orange}{16a^4b^{12}} +16b^{16} = \\ = a^{16}+ \color{blue}{8a^{12}b^4} + \color{green}{24a^8b^8} + \color{orange}{32a^4b^{12}} +16b^{16} $$$$ (a^4-2b^4)^4 = (a^4-2b^4)^2 \cdot (a^4-2b^4)^2 $$ |
| ⑤ | Find $ \left(a^4-2b^4\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^4 } $ and $ B = \color{red}{ 2b^4 }$. $$ \begin{aligned}\left(a^4-2b^4\right)^2 = \color{blue}{\left( a^4 \right)^2} -2 \cdot a^4 \cdot 2b^4 + \color{red}{\left( 2b^4 \right)^2} = a^8-4a^4b^4+4b^8\end{aligned} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{a^8-4a^4b^4+4b^8}\right) $ by each term in $ \left( a^8-4a^4b^4+4b^8\right) $. $$ \left( \color{blue}{a^8-4a^4b^4+4b^8}\right) \cdot \left( a^8-4a^4b^4+4b^8\right) = \\ = a^{16}-4a^{12}b^4+4a^8b^8-4a^{12}b^4+16a^8b^8-16a^4b^{12}+4a^8b^8-16a^4b^{12}+16b^{16} $$ |
| ⑦ | Combine like terms: $$ a^{16} \color{blue}{-4a^{12}b^4} + \color{red}{4a^8b^8} \color{blue}{-4a^{12}b^4} + \color{green}{16a^8b^8} \color{orange}{-16a^4b^{12}} + \color{green}{4a^8b^8} \color{orange}{-16a^4b^{12}} +16b^{16} = \\ = a^{16} \color{blue}{-8a^{12}b^4} + \color{green}{24a^8b^8} \color{orange}{-32a^4b^{12}} +16b^{16} $$ |
| ⑧ | Remove the parentheses by changing the sign of each term within them. $$ - \left( a^{16}-8a^{12}b^4+24a^8b^8-32a^4b^{12}+16b^{16} \right) = -a^{16}+8a^{12}b^4-24a^8b^8+32a^4b^{12}-16b^{16} $$ |
| ⑨ | Combine like terms: $$ \, \color{blue}{ \cancel{a^{16}}} \,+ \color{green}{8a^{12}b^4} + \, \color{orange}{ \cancel{24a^8b^8}} \,+ \color{red}{32a^4b^{12}} + \, \color{green}{ \cancel{16b^{16}}} \, \, \color{blue}{ -\cancel{a^{16}}} \,+ \color{green}{8a^{12}b^4} \, \color{orange}{ -\cancel{24a^8b^8}} \,+ \color{red}{32a^4b^{12}} \, \color{green}{ -\cancel{16b^{16}}} \, = \color{green}{16a^{12}b^4} + \color{red}{64a^4b^{12}} $$ |
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