L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{22}{3}+\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{63}+\left(6\right)\sqrt{175}\right)-\left(\dfrac{16}{5}+\left(-\dfrac{22}{5}\right)\sqrt{175}+\left(5\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\) et \( Y=\left(\dfrac{7}{3}\right)\sqrt{28}\) . 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{22}{3}+\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{63}+\left(6\right)\sqrt{175}\right)-\left(\dfrac{16}{5}+\left(-\dfrac{22}{5}\right)\sqrt{175}+\left(5\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\right)+\left(\left(\dfrac{7}{3}\right)\sqrt{28}\right)\\
&=&\left(\left(-\dfrac{22}{3}+\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{45}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{8}{5}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{165}{7}\right)\sqrt{7}+\left(30\right)\sqrt{7}\right)-\left(\dfrac{16}{5}+\left(-22\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)\right)+\left(\left(\dfrac{14}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{22}{3}+\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{45}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{8}{5}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{165}{7}\right)\sqrt{7}+\left(30\right)\sqrt{7}\right)-\left(\dfrac{16}{5}+\left(-22\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)+\left(\dfrac{14}{3}\right)\sqrt{7}\\
&=&-\dfrac{158}{15}+\left(-\dfrac{1202}{105}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{22}{3}+\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{63}+\left(6\right)\sqrt{175}\right)-\left(\dfrac{16}{5}+\left(-\dfrac{22}{5}\right)\sqrt{175}+\left(5\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\right)-\left(\left(\dfrac{7}{3}\right)\sqrt{28}\right)\\
&=&\left(\left(-\dfrac{22}{3}+\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{45}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{8}{5}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{165}{7}\right)\sqrt{7}+\left(30\right)\sqrt{7}\right)-\left(\dfrac{16}{5}+\left(-22\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{158}{15}+\left(-\dfrac{564}{35}\right)\sqrt{7}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{7}\right)\\
&=&-\dfrac{158}{15}+\left(-\dfrac{564}{35}\right)\sqrt{7}+\left(-\dfrac{14}{3}\right)\sqrt{7}\\
&=&-\dfrac{158}{15}+\left(-\dfrac{2182}{105}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{22}{3}+\left(\dfrac{24}{7}\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{9}{7}\right)\sqrt{175}\right)-\left(\left(\dfrac{4}{5}\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{63}+\left(6\right)\sqrt{175}\right)-\left(\dfrac{16}{5}+\left(-\dfrac{22}{5}\right)\sqrt{175}+\left(5\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\right)\times\left(\left(\dfrac{7}{3}\right)\sqrt{28}\right)\\
&=&\left(\left(-\dfrac{22}{3}+\left(\dfrac{72}{7}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{45}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{8}{5}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{165}{7}\right)\sqrt{7}+\left(30\right)\sqrt{7}\right)-\left(\dfrac{16}{5}+\left(-22\right)\sqrt{7}+\left(25\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)\right)\times\left(\left(\dfrac{14}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{158}{15}+\left(-\dfrac{564}{35}\right)\sqrt{7}\right)\left(\left(\dfrac{14}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{2212}{45}\right)\sqrt{7}+\left(-\dfrac{376}{5}\right)\sqrt{49}\\
&=&\left(-\dfrac{2212}{45}\right)\sqrt{7}-\dfrac{2632}{5}\\
\end{eqnarray*}