L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{46}{9}\right)\sqrt{50}\) et \( Y=\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{8}+\left(\dfrac{80}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}-6+\dfrac{66}{5}+\left(6\right)\sqrt{18}+\left(\dfrac{15}{2}\right)\sqrt{50}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{46}{9}\right)\sqrt{50}\right)+\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{8}+\left(\dfrac{80}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}-6+\dfrac{66}{5}+\left(6\right)\sqrt{18}+\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\dfrac{230}{9}\right)\sqrt{2}\right)+\left(\dfrac{72}{7}-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{2}+\left(\dfrac{240}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}-6+\dfrac{66}{5}+\left(18\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{230}{9}\right)\sqrt{2}+\dfrac{72}{7}-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{2}+\left(\dfrac{240}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}-6+\dfrac{66}{5}+\left(18\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{7841}{126}\right)\sqrt{2}-\dfrac{218}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{46}{9}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{8}+\left(\dfrac{80}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}-6+\dfrac{66}{5}+\left(6\right)\sqrt{18}+\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\dfrac{230}{9}\right)\sqrt{2}\right)-\left(\dfrac{72}{7}-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{2}+\left(\dfrac{240}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}-6+\dfrac{66}{5}+\left(18\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{230}{9}\right)\sqrt{2}\right)-\left(-\dfrac{218}{315}+\left(-\dfrac{1229}{14}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{230}{9}\right)\sqrt{2}+\dfrac{218}{315}+\left(\dfrac{1229}{14}\right)\sqrt{2}\\
&=&\left(\dfrac{14281}{126}\right)\sqrt{2}+\dfrac{218}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{46}{9}\right)\sqrt{50}\right)\times\left(\left(\left(\dfrac{36}{7}\right)\sqrt{4}\right)-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{8}+\left(\dfrac{80}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}-6+\dfrac{66}{5}+\left(6\right)\sqrt{18}+\left(\dfrac{15}{2}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\dfrac{230}{9}\right)\sqrt{2}\right)\times\left(\dfrac{72}{7}-\left(\dfrac{34}{9}-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{2}+\left(\dfrac{240}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}-6+\dfrac{66}{5}+\left(18\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{230}{9}\right)\sqrt{2}\right)\left(-\dfrac{218}{315}+\left(-\dfrac{1229}{14}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{10028}{567}\right)\sqrt{2}+\left(-\dfrac{141335}{63}\right)\sqrt{4}\\
&=&\left(-\dfrac{10028}{567}\right)\sqrt{2}-\dfrac{282670}{63}\\
\end{eqnarray*}