L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-1\right)\sqrt{28}+\left(9\right)\sqrt{28}\right)-\left(-\dfrac{5}{4}-\left(\left(-3\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{50}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{63}\right)\right)\) et \( Y=\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{49}+\left(-8\right)\sqrt{28}+\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{3}{5}\right)\sqrt{175}-\dfrac{23}{2}\) . 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(-1\right)\sqrt{28}+\left(9\right)\sqrt{28}\right)-\left(-\dfrac{5}{4}-\left(\left(-3\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{50}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{63}\right)\right)\right)+\left(\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{49}+\left(-8\right)\sqrt{28}+\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{3}{5}\right)\sqrt{175}-\dfrac{23}{2}\right)\\
&=&\left(\left(\left(-2\right)\sqrt{7}+\left(18\right)\sqrt{7}\right)-\left(-\dfrac{5}{4}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{250}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{231}{4}\right)\sqrt{7}\right)\right)\right)+\left(\dfrac{119}{9}-\dfrac{539}{5}+\left(-16\right)\sqrt{7}+\dfrac{119}{9}+\left(-3\right)\sqrt{7}-\dfrac{23}{2}\right)\\
&=&\left(\left(-2\right)\sqrt{7}+\left(18\right)\sqrt{7}\right)-\left(-\dfrac{5}{4}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{250}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{231}{4}\right)\sqrt{7}\right)\right)+\dfrac{119}{9}-\dfrac{539}{5}+\left(-16\right)\sqrt{7}+\dfrac{119}{9}+\left(-3\right)\sqrt{7}-\dfrac{23}{2}\\
&=&\left(\dfrac{247}{56}\right)\sqrt{7}-\dfrac{16489}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-1\right)\sqrt{28}+\left(9\right)\sqrt{28}\right)-\left(-\dfrac{5}{4}-\left(\left(-3\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{50}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{63}\right)\right)\right)-\left(\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{49}+\left(-8\right)\sqrt{28}+\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{3}{5}\right)\sqrt{175}-\dfrac{23}{2}\right)\\
&=&\left(\left(\left(-2\right)\sqrt{7}+\left(18\right)\sqrt{7}\right)-\left(-\dfrac{5}{4}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{250}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{231}{4}\right)\sqrt{7}\right)\right)\right)-\left(\dfrac{119}{9}-\dfrac{539}{5}+\left(-16\right)\sqrt{7}+\dfrac{119}{9}+\left(-3\right)\sqrt{7}-\dfrac{23}{2}\right)\\
&=&\left(\left(\dfrac{1311}{56}\right)\sqrt{7}+\dfrac{5}{4}\right)-\left(-\dfrac{8357}{90}+\left(-19\right)\sqrt{7}\right)\\
&=&\left(\dfrac{1311}{56}\right)\sqrt{7}+\dfrac{5}{4}+\dfrac{8357}{90}+\left(19\right)\sqrt{7}\\
&=&\left(\dfrac{2375}{56}\right)\sqrt{7}+\dfrac{16939}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-1\right)\sqrt{28}+\left(9\right)\sqrt{28}\right)-\left(-\dfrac{5}{4}-\left(\left(-3\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{63}\right)\right)-\left(\left(\left(\dfrac{50}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{77}{4}\right)\sqrt{63}\right)\right)\right)\times\left(\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{49}+\left(-8\right)\sqrt{28}+\left(\dfrac{17}{9}\right)\sqrt{49}+\left(-\dfrac{3}{5}\right)\sqrt{175}-\dfrac{23}{2}\right)\\
&=&\left(\left(\left(-2\right)\sqrt{7}+\left(18\right)\sqrt{7}\right)-\left(-\dfrac{5}{4}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{8}\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{250}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{231}{4}\right)\sqrt{7}\right)\right)\right)\times\left(\dfrac{119}{9}-\dfrac{539}{5}+\left(-16\right)\sqrt{7}+\dfrac{119}{9}+\left(-3\right)\sqrt{7}-\dfrac{23}{2}\right)\\
&=&\left(\left(\dfrac{1311}{56}\right)\sqrt{7}+\dfrac{5}{4}\right)\left(-\dfrac{8357}{90}+\left(-19\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{3691909}{1680}\right)\sqrt{7}+\left(-\dfrac{24909}{56}\right)\sqrt{49}-\dfrac{8357}{72}\\
&=&\left(-\dfrac{3691909}{1680}\right)\sqrt{7}-\dfrac{116269}{36}\\
\end{eqnarray*}