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