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(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\left(\left(\left(-3\right)\sqrt{125}\right)-\dfrac{12}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{45}\right)-\left(0-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)\right)-\left(\left(5\right)\sqrt{20}\right)$ et $ Y=\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(\dfrac{81}{5}\right)\sqrt{45}+\left(\left(-\dfrac{25}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{45}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{9}{4}\right)\sqrt{125}\right)+\left(-\dfrac{38}{3}\right)\sqrt{125}+\dfrac{61}{7}+\left(-\dfrac{9}{4}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+\dfrac{25}{6}+\left(\dfrac{11}{2}\right)\sqrt{20}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\left(\left(\left(-3\right)\sqrt{125}\right)-\dfrac{12}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{45}\right)-\left(0-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)\right)-\left(\left(5\right)\sqrt{20}\right)\right)+\left(\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(\dfrac{81}{5}\right)\sqrt{45}+\left(\left(-\dfrac{25}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{45}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{9}{4}\right)\sqrt{125}\right)+\left(-\dfrac{38}{3}\right)\sqrt{125}+\dfrac{61}{7}+\left(-\dfrac{9}{4}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+\dfrac{25}{6}+\left(\dfrac{11}{2}\right)\sqrt{20}\right)\\ &=&\left(\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\left(\left(\left(-15\right)\sqrt{5}\right)-\dfrac{12}{7}-\left(\left(-50\right)\sqrt{5}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)\right)-\left(\left(73\right)\sqrt{5}\right)-\left(0-\left(\left(-50\right)\sqrt{5}\right)\right)-\left(\left(10\right)\sqrt{5}\right)\right)+\left(-\dfrac{185}{9}+\left(\dfrac{243}{5}\right)\sqrt{5}-\dfrac{125}{9}-\left(\left(-\dfrac{34}{3}\right)\sqrt{5}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{45}{4}\right)\sqrt{5}\right)+\left(-\dfrac{190}{3}\right)\sqrt{5}+\dfrac{61}{7}+\left(-\dfrac{45}{4}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+\dfrac{25}{6}+\left(11\right)\sqrt{5}\right)\\ &=&\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\left(\left(\left(-15\right)\sqrt{5}\right)-\dfrac{12}{7}-\left(\left(-50\right)\sqrt{5}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)\right)-\left(\left(73\right)\sqrt{5}\right)-\left(0-\left(\left(-50\right)\sqrt{5}\right)\right)-\left(\left(10\right)\sqrt{5}\right)-\dfrac{185}{9}+\left(\dfrac{243}{5}\right)\sqrt{5}-\dfrac{125}{9}-\left(\left(-\dfrac{34}{3}\right)\sqrt{5}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{45}{4}\right)\sqrt{5}\right)+\left(-\dfrac{190}{3}\right)\sqrt{5}+\dfrac{61}{7}+\left(-\dfrac{45}{4}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+\dfrac{25}{6}+\left(11\right)\sqrt{5}\\ &=&\left(-\dfrac{6133}{30}\right)\sqrt{5}-\dfrac{3679}{252}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\left(\left(\left(-3\right)\sqrt{125}\right)-\dfrac{12}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{45}\right)-\left(0-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)\right)-\left(\left(5\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(\dfrac{81}{5}\right)\sqrt{45}+\left(\left(-\dfrac{25}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{45}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{9}{4}\right)\sqrt{125}\right)+\left(-\dfrac{38}{3}\right)\sqrt{125}+\dfrac{61}{7}+\left(-\dfrac{9}{4}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+\dfrac{25}{6}+\left(\dfrac{11}{2}\right)\sqrt{20}\right)\\ &=&\left(\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\left(\left(\left(-15\right)\sqrt{5}\right)-\dfrac{12}{7}-\left(\left(-50\right)\sqrt{5}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)\right)-\left(\left(73\right)\sqrt{5}\right)-\left(0-\left(\left(-50\right)\sqrt{5}\right)\right)-\left(\left(10\right)\sqrt{5}\right)\right)-\left(-\dfrac{185}{9}+\left(\dfrac{243}{5}\right)\sqrt{5}-\dfrac{125}{9}-\left(\left(-\dfrac{34}{3}\right)\sqrt{5}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{45}{4}\right)\sqrt{5}\right)+\left(-\dfrac{190}{3}\right)\sqrt{5}+\dfrac{61}{7}+\left(-\dfrac{45}{4}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+\dfrac{25}{6}+\left(11\right)\sqrt{5}\right)\\ &=&\left(\left(-\dfrac{3518}{15}\right)\sqrt{5}+\dfrac{12}{7}\right)-\left(-\dfrac{4111}{252}+\left(\dfrac{301}{10}\right)\sqrt{5}\right)\\ &=&\left(-\dfrac{3518}{15}\right)\sqrt{5}+\dfrac{12}{7}+\dfrac{4111}{252}+\left(-\dfrac{301}{10}\right)\sqrt{5}\\ &=&\left(-\dfrac{7939}{30}\right)\sqrt{5}+\dfrac{649}{36}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{58}{5}\right)\sqrt{20}\right)-\left(\left(\left(-3\right)\sqrt{125}\right)-\dfrac{12}{7}-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{73}{3}\right)\sqrt{45}\right)-\left(0-\left(\left(-\dfrac{50}{3}\right)\sqrt{45}\right)\right)-\left(\left(5\right)\sqrt{20}\right)\right)\times\left(\left(-\dfrac{37}{9}\right)\sqrt{25}+\left(\dfrac{81}{5}\right)\sqrt{45}+\left(\left(-\dfrac{25}{9}\right)\sqrt{25}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{45}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{9}{4}\right)\sqrt{125}\right)+\left(-\dfrac{38}{3}\right)\sqrt{125}+\dfrac{61}{7}+\left(-\dfrac{9}{4}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+\dfrac{25}{6}+\left(\dfrac{11}{2}\right)\sqrt{20}\right)\\ &=&\left(\left(\left(-\dfrac{116}{5}\right)\sqrt{5}\right)-\left(\left(\left(-15\right)\sqrt{5}\right)-\dfrac{12}{7}-\left(\left(-50\right)\sqrt{5}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)\right)-\left(\left(73\right)\sqrt{5}\right)-\left(0-\left(\left(-50\right)\sqrt{5}\right)\right)-\left(\left(10\right)\sqrt{5}\right)\right)\times\left(-\dfrac{185}{9}+\left(\dfrac{243}{5}\right)\sqrt{5}-\dfrac{125}{9}-\left(\left(-\dfrac{34}{3}\right)\sqrt{5}\right)+4+\dfrac{5}{4}-\left(\left(-\dfrac{45}{4}\right)\sqrt{5}\right)+\left(-\dfrac{190}{3}\right)\sqrt{5}+\dfrac{61}{7}+\left(-\dfrac{45}{4}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+\dfrac{25}{6}+\left(11\right)\sqrt{5}\right)\\ &=&\left(\left(-\dfrac{3518}{15}\right)\sqrt{5}+\dfrac{12}{7}\right)\left(-\dfrac{4111}{252}+\left(\dfrac{301}{10}\right)\sqrt{5}\right)\\ &=&\left(\dfrac{7328773}{1890}\right)\sqrt{5}+\left(-\dfrac{529459}{75}\right)\sqrt{25}-\dfrac{4111}{147}\\ &=&\left(\dfrac{7328773}{1890}\right)\sqrt{5}-\dfrac{8654682}{245}\\ \end{eqnarray*}