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