Ataraxy

\( %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 ; #2\right]\!\right]} \newcommand{\intEFO}[2]{\left[\!\left[#1 ; #2\right[\!\right[} \newcommand{\intEOF}[2]{\left]\!\left]#1 ; #2\right]\!\right]} \newcommand{\intEOO}[2]{\left]\!\left]#1 ; #2\right[\!\right[} \newcommand{\ou}{\vee} \newcommand{\et}{\wedge} \newcommand{\non}{\neg} \newcommand{\implique}{\Rightarrow} \newcommand{\equivalent}{\Leftrightarrow} \newcommand{\Ab}{\overline{A}} \newcommand{\Bb}{\overline{B}} \newcommand{\Cb}{\overline{C}} \newcommand{\Cl}{\texttt{Cl}} \newcommand{\ab}{\overline{a}} \newcommand{\bb}{\overline{b}} \newcommand{\cb}{\overline{c}} \newcommand{\Rel}{\mathcal{R}} \newcommand{\superepsilon}{\varepsilon\!\!\varepsilon} \newcommand{\supere}{e\!\!e} \makeatletter \newenvironment{console}{\noindent\color{white}\begin{lrbox}{\@tempboxa}\begin{minipage}{\columnwidth} \ttfamily \bfseries\vspace*{0.5cm}} {\vspace*{0.5cm}\end{minipage}\end{lrbox}\colorbox{black}{\usebox{\@tempboxa}} } \makeatother \def\ie{\textit{i.e. }} \def\cf{\textit{c.f. }} \def\vide{ { $ {\text{ }} $ } } %Commande pour les vecteurs \newcommand{\grad}{\overrightarrow{Grad}} \newcommand{\Vv}{\overrightarrow{v}} \newcommand{\Vu}{\overrightarrow{u}} \newcommand{\Vw}{\overrightarrow{w}} \newcommand{\Vup}{\overrightarrow{u'}} \newcommand{\Zero}{\overrightarrow{0}} \newcommand{\Vx}{\overrightarrow{x}} \newcommand{\Vy}{\overrightarrow{y}} \newcommand{\Vz}{\overrightarrow{z}} \newcommand{\Vt}{\overrightarrow{t}} \newcommand{\Va}{\overrightarrow{a}} \newcommand{\Vb}{\overrightarrow{b}} \newcommand{\Vc}{\overrightarrow{c}} \newcommand{\Vd}{\overrightarrow{d}} \newcommand{\Ve}[1]{\overrightarrow{e_{#1}}} \newcommand{\Vf}[1]{\overrightarrow{f_{#1}}} \newcommand{\Vn}{\overrightarrow{0}} \newcommand{\Mat}{Mat} \newcommand{\Pass}{Pass} \newcommand{\mkF}{\mathfrak{F}} \renewcommand{\sp}{Sp} \newcommand{\Co}{Co} \newcommand{\vect}[1]{\texttt{Vect}\dpl{\left( #1\right)}} \newcommand{\prodscal}[2]{\dpl{\left\langle #1\left|\vphantom{#1 #2}\right. #2\right\rangle}} \newcommand{\trans}[1]{{\vphantom{#1}}^{t}{#1}} \newcommand{\ortho}[1]{{#1}^{\bot}} \newcommand{\oplusbot}{\overset{\bot}{\oplus}} \SelectTips{cm}{12}%Change le bout des flèches dans un xymatrix \newcommand{\pourDES}[8]{ \begin{itemize} \item Pour la ligne : le premier et dernier caractère forment $#1#2$ soit $#4$ en base 10. \item Pour la colonne : les autres caractères du bloc forment $#3$ soit $#5$ en base 10. \item A l'intersection de la ligne $#4+1$ et de la colonne $#5+1$ de $S_{#8}$ se trouve l'entier $#6$ qui, codé sur $4$ bits, est \textbf{\texttt{$#7$}}. \end{itemize} } \)

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

Soit $ X=\left(\dfrac{43}{2}\right)\sqrt{4}+\left(-\dfrac{17}{3}\right)\sqrt{4}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{9}{5}\right)\sqrt{18}+\dfrac{1}{4}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{8}{3}\right)\sqrt{18}$ et $ Y=\left(\left(\dfrac{65}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{18}\right)+\left(-\dfrac{29}{9}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(\dfrac{10}{9}\right)\sqrt{4}-\dfrac{35}{8}+\left(-4\right)\sqrt{50}-6+\dfrac{19}{2}+\left(\left(\dfrac{29}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{4}\right)+\dfrac{79}{4}-\left(\left(-\dfrac{3}{4}\right)\sqrt{8}\right)$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\left(\dfrac{43}{2}\right)\sqrt{4}+\left(-\dfrac{17}{3}\right)\sqrt{4}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{9}{5}\right)\sqrt{18}+\dfrac{1}{4}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{8}{3}\right)\sqrt{18}\right)+\left(\left(\left(\dfrac{65}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{18}\right)+\left(-\dfrac{29}{9}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(\dfrac{10}{9}\right)\sqrt{4}-\dfrac{35}{8}+\left(-4\right)\sqrt{50}-6+\dfrac{19}{2}+\left(\left(\dfrac{29}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{4}\right)+\dfrac{79}{4}-\left(\left(-\dfrac{3}{4}\right)\sqrt{8}\right)\right)\\ &=&\left(43-\dfrac{34}{3}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{27}{5}\right)\sqrt{2}+\dfrac{1}{4}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-8\right)\sqrt{2}\right)+\left(\left(\left(\dfrac{195}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{2}\right)+\left(-\dfrac{29}{3}\right)\sqrt{2}-\dfrac{8}{3}+\dfrac{20}{9}-\dfrac{35}{8}+\left(-20\right)\sqrt{2}-6+\dfrac{19}{2}+\dfrac{29}{3}+\dfrac{35}{4}+\dfrac{79}{4}-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)\right)\\ &=&43-\dfrac{34}{3}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{27}{5}\right)\sqrt{2}+\dfrac{1}{4}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-8\right)\sqrt{2}+\left(\left(\dfrac{195}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{2}\right)+\left(-\dfrac{29}{3}\right)\sqrt{2}-\dfrac{8}{3}+\dfrac{20}{9}-\dfrac{35}{8}+\left(-20\right)\sqrt{2}-6+\dfrac{19}{2}+\dfrac{29}{3}+\dfrac{35}{4}+\dfrac{79}{4}-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)\\ &=&\dfrac{30145}{504}+\left(\dfrac{291}{40}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\dfrac{43}{2}\right)\sqrt{4}+\left(-\dfrac{17}{3}\right)\sqrt{4}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{9}{5}\right)\sqrt{18}+\dfrac{1}{4}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{8}{3}\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{65}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{18}\right)+\left(-\dfrac{29}{9}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(\dfrac{10}{9}\right)\sqrt{4}-\dfrac{35}{8}+\left(-4\right)\sqrt{50}-6+\dfrac{19}{2}+\left(\left(\dfrac{29}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{4}\right)+\dfrac{79}{4}-\left(\left(-\dfrac{3}{4}\right)\sqrt{8}\right)\right)\\ &=&\left(43-\dfrac{34}{3}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{27}{5}\right)\sqrt{2}+\dfrac{1}{4}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-8\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{195}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{2}\right)+\left(-\dfrac{29}{3}\right)\sqrt{2}-\dfrac{8}{3}+\dfrac{20}{9}-\dfrac{35}{8}+\left(-20\right)\sqrt{2}-6+\dfrac{19}{2}+\dfrac{29}{3}+\dfrac{35}{4}+\dfrac{79}{4}-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)\right)\\ &=&\left(\dfrac{643}{28}+\left(-\dfrac{311}{10}\right)\sqrt{2}\right)-\left(\left(\dfrac{307}{8}\right)\sqrt{2}+\dfrac{2653}{72}\right)\\ &=&\dfrac{643}{28}+\left(-\dfrac{311}{10}\right)\sqrt{2}+\left(-\dfrac{307}{8}\right)\sqrt{2}-\dfrac{2653}{72}\\ &=&-\dfrac{6997}{504}+\left(-\dfrac{2779}{40}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\dfrac{43}{2}\right)\sqrt{4}+\left(-\dfrac{17}{3}\right)\sqrt{4}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{9}{5}\right)\sqrt{18}+\dfrac{1}{4}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{19}{4}\right)\sqrt{18}+\left(-\dfrac{8}{3}\right)\sqrt{18}\right)\times\left(\left(\left(\dfrac{65}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{18}\right)+\left(-\dfrac{29}{9}\right)\sqrt{18}+\left(-\dfrac{4}{3}\right)\sqrt{4}+\left(\dfrac{10}{9}\right)\sqrt{4}-\dfrac{35}{8}+\left(-4\right)\sqrt{50}-6+\dfrac{19}{2}+\left(\left(\dfrac{29}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{4}\right)+\dfrac{79}{4}-\left(\left(-\dfrac{3}{4}\right)\sqrt{8}\right)\right)\\ &=&\left(43-\dfrac{34}{3}-\dfrac{37}{7}-\dfrac{29}{3}+6+\left(\dfrac{27}{5}\right)\sqrt{2}+\dfrac{1}{4}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-\dfrac{57}{4}\right)\sqrt{2}+\left(-8\right)\sqrt{2}\right)\times\left(\left(\left(\dfrac{195}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{2}\right)+\left(-\dfrac{29}{3}\right)\sqrt{2}-\dfrac{8}{3}+\dfrac{20}{9}-\dfrac{35}{8}+\left(-20\right)\sqrt{2}-6+\dfrac{19}{2}+\dfrac{29}{3}+\dfrac{35}{4}+\dfrac{79}{4}-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)\right)\\ &=&\left(\dfrac{643}{28}+\left(-\dfrac{311}{10}\right)\sqrt{2}\right)\left(\left(\dfrac{307}{8}\right)\sqrt{2}+\dfrac{2653}{72}\right)\\ &=&\left(-\dfrac{2668117}{10080}\right)\sqrt{2}+\dfrac{243697}{288}+\left(-\dfrac{95477}{80}\right)\sqrt{4}\\ &=&\left(-\dfrac{2668117}{10080}\right)\sqrt{2}-\dfrac{2218687}{1440}\\ \end{eqnarray*}