Answer
$$(x+x^2+x^6+x^6)^3=8x^{18}+12x^{14}+12x^{13}+6x^{10}+12x^9+6x^8+x^6+3x^5+3x^4+x^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+x^2+x^6+x^6)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(2x^6+x^2+x)^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}8x^{18}+12x^{14}+12x^{13}+6x^{10}+12x^9+6x^8+x^6+3x^5+3x^4+x^3\end{aligned} $$ | |
| ① | Combine like terms: $$ x+x^2+ \color{blue}{x^6} + \color{blue}{x^6} = \color{blue}{2x^6} +x^2+x $$ |
| ② | Multiply each term of $ \left( \color{blue}{2x^6+x^2+x}\right) $ by each term in $ \left( 2x^6+x^2+x\right) $. $$ \left( \color{blue}{2x^6+x^2+x}\right) \cdot \left( 2x^6+x^2+x\right) = 4x^{12}+2x^8+2x^7+2x^8+x^4+x^3+2x^7+x^3+x^2 $$ |
| ③ | Combine like terms: $$ 4x^{12}+ \color{blue}{2x^8} + \color{red}{2x^7} + \color{blue}{2x^8} +x^4+ \color{green}{x^3} + \color{red}{2x^7} + \color{green}{x^3} +x^2 = \\ = 4x^{12}+ \color{blue}{4x^8} + \color{red}{4x^7} +x^4+ \color{green}{2x^3} +x^2 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{4x^{12}+4x^8+4x^7+x^4+2x^3+x^2}\right) $ by each term in $ \left( 2x^6+x^2+x\right) $. $$ \left( \color{blue}{4x^{12}+4x^8+4x^7+x^4+2x^3+x^2}\right) \cdot \left( 2x^6+x^2+x\right) = \\ = 8x^{18}+4x^{14}+4x^{13}+8x^{14}+4x^{10}+4x^9+8x^{13}+4x^9+4x^8+2x^{10}+x^6+x^5+4x^9+2x^5+2x^4+2x^8+x^4+x^3 $$ |
| ⑤ | Combine like terms: $$ 8x^{18}+ \color{blue}{4x^{14}} + \color{red}{4x^{13}} + \color{blue}{8x^{14}} + \color{green}{4x^{10}} + \color{orange}{4x^9} + \color{red}{8x^{13}} + \color{blue}{4x^9} + \color{red}{4x^8} + \color{green}{2x^{10}} +x^6+ \color{green}{x^5} + \color{blue}{4x^9} + \color{green}{2x^5} + \color{orange}{2x^4} + \color{red}{2x^8} + \color{orange}{x^4} +x^3 = \\ = 8x^{18}+ \color{blue}{12x^{14}} + \color{red}{12x^{13}} + \color{green}{6x^{10}} + \color{blue}{12x^9} + \color{red}{6x^8} +x^6+ \color{green}{3x^5} + \color{orange}{3x^4} +x^3 $$ |
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