L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{69}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{4}\right)-\dfrac{61}{3}-\left(\left(-\dfrac{15}{4}\right)\sqrt{8}\right)\) et \( Y=\left(-\dfrac{29}{8}\right)\sqrt{8}\) . 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{69}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{4}\right)-\dfrac{61}{3}-\left(\left(-\dfrac{15}{4}\right)\sqrt{8}\right)\right)+\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{345}{4}\right)\sqrt{2}\right)-\dfrac{22}{3}-\dfrac{5}{2}-\dfrac{61}{3}-\left(\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\right)+\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{345}{4}\right)\sqrt{2}\right)-\dfrac{22}{3}-\dfrac{5}{2}-\dfrac{61}{3}-\left(\left(-\dfrac{15}{2}\right)\sqrt{2}\right)+\left(-\dfrac{29}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{173}{2}\right)\sqrt{2}-\dfrac{181}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{69}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{4}\right)-\dfrac{61}{3}-\left(\left(-\dfrac{15}{4}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{345}{4}\right)\sqrt{2}\right)-\dfrac{22}{3}-\dfrac{5}{2}-\dfrac{61}{3}-\left(\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{2}-\dfrac{181}{6}\right)-\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{375}{4}\right)\sqrt{2}-\dfrac{181}{6}+\left(\dfrac{29}{4}\right)\sqrt{2}\\
&=&\left(101\right)\sqrt{2}-\dfrac{181}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{69}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{5}{4}\right)\sqrt{4}\right)-\dfrac{61}{3}-\left(\left(-\dfrac{15}{4}\right)\sqrt{8}\right)\right)\times\left(\left(-\dfrac{29}{8}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{345}{4}\right)\sqrt{2}\right)-\dfrac{22}{3}-\dfrac{5}{2}-\dfrac{61}{3}-\left(\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\right)\times\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{2}-\dfrac{181}{6}\right)\left(\left(-\dfrac{29}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{10875}{16}\right)\sqrt{4}+\left(\dfrac{5249}{24}\right)\sqrt{2}\\
&=&-\dfrac{10875}{8}+\left(\dfrac{5249}{24}\right)\sqrt{2}\\
\end{eqnarray*}