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