L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{77}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{14}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{5}{6}\right)\sqrt{9}\right)-\dfrac{21}{8}\) et \( Y=\left(-\dfrac{7}{4}\right)\sqrt{9}\) . 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(\left(\dfrac{77}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{14}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{5}{6}\right)\sqrt{9}\right)-\dfrac{21}{8}\right)+\left(\left(-\dfrac{7}{4}\right)\sqrt{9}\right)\\
&=&\left(\left(\left(\left(\dfrac{77}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{42}{5}\right)\sqrt{3}\right)-\dfrac{5}{2}-\dfrac{21}{8}\right)+\left(-\dfrac{21}{4}\right)\\
&=&\left(\left(\left(\dfrac{77}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{42}{5}\right)\sqrt{3}\right)-\dfrac{5}{2}-\dfrac{21}{8}-\dfrac{21}{4}\\
&=&\left(-\dfrac{1593}{70}\right)\sqrt{3}-\dfrac{221}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{77}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{14}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{5}{6}\right)\sqrt{9}\right)-\dfrac{21}{8}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{9}\right)\\
&=&\left(\left(\left(\left(\dfrac{77}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{42}{5}\right)\sqrt{3}\right)-\dfrac{5}{2}-\dfrac{21}{8}\right)-\left(-\dfrac{21}{4}\right)\\
&=&\left(\left(-\dfrac{1593}{70}\right)\sqrt{3}-\dfrac{79}{6}\right)-\left(-\dfrac{21}{4}\right)\\
&=&\left(-\dfrac{1593}{70}\right)\sqrt{3}-\dfrac{79}{6}+\dfrac{21}{4}\\
&=&\left(-\dfrac{1593}{70}\right)\sqrt{3}-\dfrac{95}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{77}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{74}{7}\right)\sqrt{75}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{14}{5}\right)\sqrt{27}\right)-\left(\left(\dfrac{5}{6}\right)\sqrt{9}\right)-\dfrac{21}{8}\right)\times\left(\left(-\dfrac{7}{4}\right)\sqrt{9}\right)\\
&=&\left(\left(\left(\left(\dfrac{77}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{370}{7}\right)\sqrt{3}\right)-\dfrac{50}{3}\right)+\dfrac{69}{8}-\left(\left(\dfrac{42}{5}\right)\sqrt{3}\right)-\dfrac{5}{2}-\dfrac{21}{8}\right)\times\left(-\dfrac{21}{4}\right)\\
&=&\left(\left(-\dfrac{1593}{70}\right)\sqrt{3}-\dfrac{79}{6}\right)\left(-\dfrac{21}{4}\right)\\
&=&\left(\dfrac{4779}{40}\right)\sqrt{3}+\dfrac{553}{8}\\
&=&\left(\dfrac{4779}{40}\right)\sqrt{3}+\dfrac{553}{8}\\
\end{eqnarray*}