\( %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Mes commandes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\multirows}[3]{\multirow{#1}{#2}{$#3$}}%pour rester en mode math \renewcommand{\arraystretch}{1.3}%pour augmenter la taille des case \newcommand{\point}[1]{\marginnote{\small\vspace*{-1em} #1}}%pour indiquer les points ou le temps \newcommand{\dpl}[1]{\displaystyle{#1}}%megamode \newcommand{\A}{\mathscr{A}} \newcommand{\LN}{\mathscr{N}} \newcommand{\LL}{\mathscr{L}} \newcommand{\K}{\mathbb{K}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\M}{\mathcal{M}} \newcommand{\D}{\mathbb{D}} \newcommand{\E}{\mathcal{E}} \renewcommand{\P}{\mathcal{P}} \newcommand{\G}{\mathcal{G}} \newcommand{\Kk}{\mathcal{K}} \newcommand{\Cc}{\mathcal{C}} \newcommand{\Zz}{\mathcal{Z}} \newcommand{\Ss}{\mathcal{S}} \newcommand{\B}{\mathbb{B}} \newcommand{\inde}{\bot\!\!\!\bot} \newcommand{\Proba}{\mathbb{P}} \newcommand{\Esp}[1]{\dpl{\mathbb{E}\left(#1\right)}} \newcommand{\Var}[1]{\dpl{\mathbb{V}\left(#1\right)}} \newcommand{\Cov}[1]{\dpl{Cov\left(#1\right)}} \newcommand{\base}{\mathcal{B}} \newcommand{\Som}{\textbf{Som}} \newcommand{\Chain}{\textbf{Chain}} \newcommand{\Ar}{\textbf{Ar}} \newcommand{\Arc}{\textbf{Arc}} \newcommand{\Min}{\text{Min}} \newcommand{\Max}{\text{Max}} \newcommand{\Ker}{\text{Ker}} \renewcommand{\Im}{\text{Im}} \newcommand{\Sup}{\text{Sup}} \newcommand{\Inf}{\text{Inf}} \renewcommand{\det}{\texttt{det}} \newcommand{\GL}{\text{GL}} \newcommand{\crossmark}{\text{\ding{55}}} \renewcommand{\checkmark}{\text{\ding{51}}} \newcommand{\Card}{\sharp} \newcommand{\Surligne}[2]{\text{\colorbox{#1}{ #2 }}} \newcommand{\SurligneMM}[2]{\text{\colorbox{#1}{ #2 }}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \renewcommand{\lim}[1]{\underset{#1}{lim}\,} \newcommand{\nonor}[1]{\left|#1\right|} \newcommand{\Un}{1\!\!1} \newcommand{\sepon}{\setlength{\columnseprule}{0.5pt}} \newcommand{\sepoff}{\setlength{\columnseprule}{0pt}} \newcommand{\flux}{Flux} \newcommand{\Cpp}{\texttt{C++\ }} \newcommand{\Python}{\texttt{Python\ }} %\newcommand{\comb}[2]{\begin{pmatrix} #1\\ #2\end{pmatrix}} \newcommand{\comb}[2]{C_{#1}^{#2}} \newcommand{\arrang}[2]{A_{#1}^{#2}} \newcommand{\supp}[1]{Supp\left(#1\right)} \newcommand{\BB}{\mathcal{B}} \newcommand{\arc}[1]{\overset{\rotatebox{90}{)}}{#1}} \newcommand{\modpi}{\equiv_{2\pi}} \renewcommand{\Re}{Re} \renewcommand{\Im}{Im} \renewcommand{\bar}[1]{\overline{#1}} \newcommand{\mat}{\mathcal{M}} \newcommand{\und}[1]{{\mathbf{\color{red}\underline{#1}}}} \newcommand{\rdots}{\text{\reflectbox{$\ddots$}}} \newcommand{\Compa}{Compa} \newcommand{\dint}{\dpl{\int}} \newcommand{\intEFF}[2]{\left[\!\left[#1 ; 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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.

Exercice


Calculer \[\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-12}{10}}{6}}{7}\times\dfrac{\dfrac{7}{\dfrac{29}{11}}}{\dfrac{35}{3}}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-12}{10}}{6}}{7}\times\dfrac{\dfrac{7}{\dfrac{29}{11}}}{\dfrac{35}{3}}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}=\dfrac{4906072710}{3241843111500000}\) . Voici le détail : \begin{eqnarray*} X &=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{6}{5}}{6}}{7}\times\dfrac{\dfrac{7}{\dfrac{29}{11}}}{\dfrac{35}{3}}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{1}{5}}{7}\times\dfrac{\dfrac{7}{\dfrac{29}{11}}}{\dfrac{35}{3}}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{\dfrac{7}{\dfrac{29}{11}}}{\dfrac{35}{3}}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{\dfrac{77}{29}}{\dfrac{35}{3}}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\left(\dfrac{35}{3}-\dfrac{31}{5}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\left(\dfrac{82}{15}+8\right)-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\dfrac{202}{15}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{31}{5}\times\dfrac{29}{11}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\dfrac{202}{15}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{899}{55}\times\left(-8\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\dfrac{202}{15}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\left(-\dfrac{7192}{55}\right)\times\left(-8\right)}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\dfrac{202}{15}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{57536}{55}}{\left(-\dfrac{41}{2}\right)\times8}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{1}{35}\right)\times\dfrac{33}{145}\times\dfrac{202}{15}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{57536}{55}}{-164}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\left(-\dfrac{33}{5075}\right)\times\dfrac{202}{15}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{57536}{55}}{-164}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{2222}{25375}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}-\dfrac{\dfrac{57536}{55}}{-164}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{2222}{25375}-8-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{205222}{25375}-9+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{433597}{25375}+\dfrac{31}{5}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{\left(8-6-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{\left(2-\dfrac{35}{3}\right)+9-\left(-8+\dfrac{31}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{-\dfrac{29}{3}+9-\left(-8+\dfrac{31}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{-\dfrac{29}{3}+9-\left(\dfrac{15}{2}+\dfrac{14384}{2255}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{-\dfrac{29}{3}+9-\left(\dfrac{62593}{4510}+8\right)-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{-\dfrac{29}{3}+9-\dfrac{98673}{4510}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8+\dfrac{-\dfrac{31}{2}}{7}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{276272}{25375}}{10}}{-8}}{-8}}{-12}}{-\dfrac{29}{3}+9-\dfrac{98673}{4510}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{138136}{126875}}{-8}}{-8}}{-12}}{-\dfrac{29}{3}+9-\dfrac{98673}{4510}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{17267}{126875}}{-8}}{-12}}{-\dfrac{29}{3}+9-\dfrac{98673}{4510}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{29}{3}+9-\dfrac{98673}{4510}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{2}{3}-\dfrac{98673}{4510}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{305039}{13530}-\dfrac{35}{3}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{462889}{13530}}}{-\dfrac{31}{2}-\dfrac{5}{3}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{462889}{13530}}}{-\dfrac{103}{6}-8-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{462889}{13530}}}{-\dfrac{151}{6}-\dfrac{31}{14}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{17267}{1015000}}{-12}}{-\dfrac{462889}{13530}}}{-\dfrac{575}{21}}\\&=&\dfrac{\dfrac{\dfrac{17267}{12180000}}{-\dfrac{462889}{13530}}}{-\dfrac{575}{21}}\\&=&\dfrac{-\dfrac{233622510}{5637988020000}}{-\dfrac{575}{21}}\\&=&\dfrac{4906072710}{3241843111500000}\\ \end{eqnarray*}