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$$\underset{\,\,\,\,{\textcolor[rgb]{0.66,0.66,0.66}{\rightarrow {\mathtt{x, y}}}}}{{solve}}{\left(\begin{array}{l}\left({\mathtt{x}}{\mathtt{\,\times\,}}{\mathtt{10}}{\mathtt{\,\small\textbf+\,}}{\mathtt{y}}\right)={\mathtt{8}}{\mathtt{\,\times\,}}\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{y}}\right)\\
{\mathtt{x}}={\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}\end{array}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{7}}\\
{\mathtt{y}} = {\mathtt{2}}\\
\end{array} \right\}$$

Zahl (x*10+y) = 72

Probe:

(7+2)*8=72 ✓
7=2+5 ✓

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{8}} = {\mathtt{50}} \Rightarrow {\mathtt{x}} = {\mathtt{14}}$$

3*x+8=50
3*x+8-8=50-8
3*x=42
3*x/3=42/3
x=14

$$\underset{\,\,\,\,{\textcolor[rgb]{0.66,0.66,0.66}{\rightarrow {\mathtt{x, y}}}}}{{solve}}{\left(\begin{array}{l}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{y}}={\mathtt{25}}\\
{\mathtt{x}}=\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right){\mathtt{\,\times\,}}{\mathtt{y}}\end{array}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{{\mathtt{25}}}{{\mathtt{7}}}}\\
{\mathtt{y}} = {\frac{{\mathtt{150}}}{{\mathtt{7}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{3.571\: \!428\: \!571\: \!428\: \!571\: \!4}}\\
{\mathtt{y}} = {\mathtt{21.428\: \!571\: \!428\: \!571\: \!428\: \!6}}\\
\end{array} \right\}$$

$${\left({\mathtt{\,-\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{i}\right)}^{{\mathtt{7}}} = {\mathtt{\,-\,}}{\mathtt{8}}{\mathtt{\,-\,}}{\mathtt{8}}{i}$$

$${expand}{\left({\left({\mathtt{\,-\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{i}\right)}^{{\mathtt{7}}}\right)} = {{i}}^{{\mathtt{7}}}{\mathtt{\,-\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{{i}}^{{\mathtt{6}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}{\mathtt{\,\times\,}}{{i}}^{{\mathtt{5}}}{\mathtt{\,-\,}}{\mathtt{35}}{\mathtt{\,\times\,}}{{i}}^{{\mathtt{4}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{35}}{\mathtt{\,\times\,}}{{i}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{21}}{\mathtt{\,\times\,}}{{i}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}$$

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}} = -{\mathtt{2}} \Rightarrow {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{2}}}{{\mathtt{3}}}} \Rightarrow {\mathtt{x}} = -{\mathtt{0.666\: \!666\: \!666\: \!666\: \!666\: \!7}}$$

$${\mathtt{8}}{!} = {\mathtt{40\,320}}$$

idea: divide the shelf in 8 imaginary slots.
you can place the first book in one of 8 slots, that's 8 possiblities.

now one slot is occupied, so the shelf has only 7 remaining slots. the second book can now be placed in one of 7 slots, that's 7 possiblities.

and so on..

the last book can only be placed in excactly one remaining slot, that's 1 possibility.
total possibilities = 8*7*6*5*4*3*2*1 = 8! (factorial) = 40320

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}} = -{\mathtt{2}} \Rightarrow {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{2}}}{{\mathtt{3}}}} \Rightarrow {\mathtt{x}} = -{\mathtt{0.666\: \!666\: \!666\: \!666\: \!666\: \!7}}$$

$${factorize}{\left({\mathtt{20}}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{4}} = {{\mathtt{2}}}^{{\mathtt{2}}}\\
{\mathtt{5}}\\
\end{array} \right\}$$

hi zegroes!

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