L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{49}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}\) et \( Y=\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{28}\right)\right)-\left(\left(-2\right)\sqrt{28}+\left(\dfrac{39}{4}\right)\sqrt{63}+\left(-\dfrac{7}{9}\right)\sqrt{49}+\left(-7\right)\sqrt{175}\right)-\left(\dfrac{77}{2}+\left(4\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\dfrac{77}{8}\right)\sqrt{175}+\left(-8\right)\sqrt{175}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{49}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}\right)+\left(\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{28}\right)\right)-\left(\left(-2\right)\sqrt{28}+\left(\dfrac{39}{4}\right)\sqrt{63}+\left(-\dfrac{7}{9}\right)\sqrt{49}+\left(-7\right)\sqrt{175}\right)-\left(\dfrac{77}{2}+\left(4\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\dfrac{77}{8}\right)\sqrt{175}+\left(-8\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(\dfrac{98}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}\right)+\left(\left(\left(\left(-\dfrac{74}{5}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{7}\right)\right)-\left(\left(-4\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}-\dfrac{49}{9}+\left(-35\right)\sqrt{7}\right)-\left(\dfrac{77}{2}+28-\dfrac{329}{9}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\dfrac{385}{8}\right)\sqrt{7}+\left(-40\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{98}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}+\left(\left(\left(-\dfrac{74}{5}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{7}\right)\right)-\left(\left(-4\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}-\dfrac{49}{9}+\left(-35\right)\sqrt{7}\right)-\left(\dfrac{77}{2}+28-\dfrac{329}{9}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\dfrac{385}{8}\right)\sqrt{7}+\left(-40\right)\sqrt{7}\right)\\
&=&\left(\dfrac{2363}{35}\right)\sqrt{7}-\dfrac{49}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{49}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}\right)-\left(\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{28}\right)\right)-\left(\left(-2\right)\sqrt{28}+\left(\dfrac{39}{4}\right)\sqrt{63}+\left(-\dfrac{7}{9}\right)\sqrt{49}+\left(-7\right)\sqrt{175}\right)-\left(\dfrac{77}{2}+\left(4\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\dfrac{77}{8}\right)\sqrt{175}+\left(-8\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(\dfrac{98}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}\right)-\left(\left(\left(\left(-\dfrac{74}{5}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{7}\right)\right)-\left(\left(-4\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}-\dfrac{49}{9}+\left(-35\right)\sqrt{7}\right)-\left(\dfrac{77}{2}+28-\dfrac{329}{9}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\dfrac{385}{8}\right)\sqrt{7}+\left(-40\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{1779}{40}\right)\sqrt{7}\right)-\left(\left(\dfrac{6451}{280}\right)\sqrt{7}-\dfrac{49}{2}\right)\\
&=&\left(\dfrac{1779}{40}\right)\sqrt{7}+\left(-\dfrac{6451}{280}\right)\sqrt{7}+\dfrac{49}{2}\\
&=&\left(\dfrac{3001}{140}\right)\sqrt{7}+\dfrac{49}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{49}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{63}\right)+\left(-2\right)\sqrt{28}\right)\times\left(\left(\left(\left(-\dfrac{37}{5}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{28}\right)\right)-\left(\left(-2\right)\sqrt{28}+\left(\dfrac{39}{4}\right)\sqrt{63}+\left(-\dfrac{7}{9}\right)\sqrt{49}+\left(-7\right)\sqrt{175}\right)-\left(\dfrac{77}{2}+\left(4\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{49}\right)-\left(\left(\dfrac{19}{4}\right)\sqrt{63}+\left(\dfrac{77}{8}\right)\sqrt{175}+\left(-8\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(\dfrac{98}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{231}{8}\right)\sqrt{7}\right)+\left(-4\right)\sqrt{7}\right)\times\left(\left(\left(\left(-\dfrac{74}{5}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{63}{4}\right)\sqrt{7}\right)\right)-\left(\left(-4\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}-\dfrac{49}{9}+\left(-35\right)\sqrt{7}\right)-\left(\dfrac{77}{2}+28-\dfrac{329}{9}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{7}+\left(\dfrac{385}{8}\right)\sqrt{7}+\left(-40\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{1779}{40}\right)\sqrt{7}\right)\left(\left(\dfrac{6451}{280}\right)\sqrt{7}-\dfrac{49}{2}\right)\\
&=&\left(\dfrac{11476329}{11200}\right)\sqrt{49}+\left(-\dfrac{87171}{80}\right)\sqrt{7}\\
&=&\dfrac{11476329}{1600}+\left(-\dfrac{87171}{80}\right)\sqrt{7}\\
\end{eqnarray*}