L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{25}+\left(-\dfrac{24}{7}\right)\sqrt{125}+\left(-6\right)\sqrt{25}-1+\left(-\dfrac{19}{2}\right)\sqrt{45}+\left(-1\right)\sqrt{20}+\left(6\right)\sqrt{125}+\left(-\dfrac{4}{3}\right)\sqrt{25}-\dfrac{29}{9}+\left(\dfrac{21}{2}\right)\sqrt{125}+\left(8\right)\sqrt{45}-8+\left(-\dfrac{16}{3}\right)\sqrt{25}+\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(\dfrac{31}{4}\right)\sqrt{20}\) et \( Y=\left(\dfrac{13}{6}\right)\sqrt{20}+\left(-\dfrac{71}{2}\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{25}+\left(-\dfrac{24}{7}\right)\sqrt{125}+\left(-6\right)\sqrt{25}-1+\left(-\dfrac{19}{2}\right)\sqrt{45}+\left(-1\right)\sqrt{20}+\left(6\right)\sqrt{125}+\left(-\dfrac{4}{3}\right)\sqrt{25}-\dfrac{29}{9}+\left(\dfrac{21}{2}\right)\sqrt{125}+\left(8\right)\sqrt{45}-8+\left(-\dfrac{16}{3}\right)\sqrt{25}+\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(\dfrac{31}{4}\right)\sqrt{20}\right)+\left(\left(\dfrac{13}{6}\right)\sqrt{20}+\left(-\dfrac{71}{2}\right)\sqrt{25}\right)\\
&=&\left(\left(-\dfrac{104}{9}\right)\sqrt{5}+40+\left(-\dfrac{120}{7}\right)\sqrt{5}-30-1+\left(-\dfrac{57}{2}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(30\right)\sqrt{5}-\dfrac{20}{3}-\dfrac{29}{9}+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(24\right)\sqrt{5}-8-\dfrac{80}{3}+\left(-\dfrac{104}{9}\right)\sqrt{5}+\left(\dfrac{31}{2}\right)\sqrt{5}\right)+\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{355}{2}\right)\\
&=&\left(-\dfrac{104}{9}\right)\sqrt{5}+40+\left(-\dfrac{120}{7}\right)\sqrt{5}-30-1+\left(-\dfrac{57}{2}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(30\right)\sqrt{5}-\dfrac{20}{3}-\dfrac{29}{9}+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(24\right)\sqrt{5}-8-\dfrac{80}{3}+\left(-\dfrac{104}{9}\right)\sqrt{5}+\left(\dfrac{31}{2}\right)\sqrt{5}+\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{355}{2}\\
&=&\left(\dfrac{7003}{126}\right)\sqrt{5}-\dfrac{3835}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{25}+\left(-\dfrac{24}{7}\right)\sqrt{125}+\left(-6\right)\sqrt{25}-1+\left(-\dfrac{19}{2}\right)\sqrt{45}+\left(-1\right)\sqrt{20}+\left(6\right)\sqrt{125}+\left(-\dfrac{4}{3}\right)\sqrt{25}-\dfrac{29}{9}+\left(\dfrac{21}{2}\right)\sqrt{125}+\left(8\right)\sqrt{45}-8+\left(-\dfrac{16}{3}\right)\sqrt{25}+\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(\dfrac{31}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{13}{6}\right)\sqrt{20}+\left(-\dfrac{71}{2}\right)\sqrt{25}\right)\\
&=&\left(\left(-\dfrac{104}{9}\right)\sqrt{5}+40+\left(-\dfrac{120}{7}\right)\sqrt{5}-30-1+\left(-\dfrac{57}{2}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(30\right)\sqrt{5}-\dfrac{20}{3}-\dfrac{29}{9}+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(24\right)\sqrt{5}-8-\dfrac{80}{3}+\left(-\dfrac{104}{9}\right)\sqrt{5}+\left(\dfrac{31}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{355}{2}\right)\\
&=&\left(\left(\dfrac{6457}{126}\right)\sqrt{5}-\dfrac{320}{9}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{355}{2}\right)\\
&=&\left(\dfrac{6457}{126}\right)\sqrt{5}-\dfrac{320}{9}+\left(-\dfrac{13}{3}\right)\sqrt{5}+\dfrac{355}{2}\\
&=&\left(\dfrac{5911}{126}\right)\sqrt{5}+\dfrac{2555}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{25}+\left(-\dfrac{24}{7}\right)\sqrt{125}+\left(-6\right)\sqrt{25}-1+\left(-\dfrac{19}{2}\right)\sqrt{45}+\left(-1\right)\sqrt{20}+\left(6\right)\sqrt{125}+\left(-\dfrac{4}{3}\right)\sqrt{25}-\dfrac{29}{9}+\left(\dfrac{21}{2}\right)\sqrt{125}+\left(8\right)\sqrt{45}-8+\left(-\dfrac{16}{3}\right)\sqrt{25}+\left(-\dfrac{52}{9}\right)\sqrt{20}+\left(\dfrac{31}{4}\right)\sqrt{20}\right)\times\left(\left(\dfrac{13}{6}\right)\sqrt{20}+\left(-\dfrac{71}{2}\right)\sqrt{25}\right)\\
&=&\left(\left(-\dfrac{104}{9}\right)\sqrt{5}+40+\left(-\dfrac{120}{7}\right)\sqrt{5}-30-1+\left(-\dfrac{57}{2}\right)\sqrt{5}+\left(-2\right)\sqrt{5}+\left(30\right)\sqrt{5}-\dfrac{20}{3}-\dfrac{29}{9}+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(24\right)\sqrt{5}-8-\dfrac{80}{3}+\left(-\dfrac{104}{9}\right)\sqrt{5}+\left(\dfrac{31}{2}\right)\sqrt{5}\right)\times\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{355}{2}\right)\\
&=&\left(\left(\dfrac{6457}{126}\right)\sqrt{5}-\dfrac{320}{9}\right)\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{355}{2}\right)\\
&=&\left(\dfrac{83941}{378}\right)\sqrt{25}+\left(-\dfrac{6993185}{756}\right)\sqrt{5}+\dfrac{56800}{9}\\
&=&\dfrac{2805305}{378}+\left(-\dfrac{6993185}{756}\right)\sqrt{5}\\
\end{eqnarray*}