L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-2\right)\sqrt{175}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{28}+\left(\dfrac{1}{3}\right)\sqrt{63}-\dfrac{29}{8}\right)-\left(\left(8\right)\sqrt{28}+\left(\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}\right)\) et \( Y=\left(\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{70}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{47}{5}\right)\sqrt{63}+\left(-\dfrac{15}{2}\right)\sqrt{28}\right)\) . 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(-2\right)\sqrt{175}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{28}+\left(\dfrac{1}{3}\right)\sqrt{63}-\dfrac{29}{8}\right)-\left(\left(8\right)\sqrt{28}+\left(\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}\right)\right)+\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{70}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{47}{5}\right)\sqrt{63}+\left(-\dfrac{15}{2}\right)\sqrt{28}\right)\right)\\
&=&\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(\dfrac{76}{3}\right)\sqrt{7}+\left(1\right)\sqrt{7}-\dfrac{29}{8}\right)-\left(\left(16\right)\sqrt{7}+\left(\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}\right)\right)+\left(\left(\left(-10\right)\sqrt{7}+\left(\dfrac{140}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{141}{5}\right)\sqrt{7}+\left(-15\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-10\right)\sqrt{7}\right)-\left(\left(\dfrac{76}{3}\right)\sqrt{7}+\left(1\right)\sqrt{7}-\dfrac{29}{8}\right)-\left(\left(16\right)\sqrt{7}+\left(\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}\right)+\left(\left(-10\right)\sqrt{7}+\left(\dfrac{140}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{141}{5}\right)\sqrt{7}+\left(-15\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{476}{5}\right)\sqrt{7}+\dfrac{29}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-2\right)\sqrt{175}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{28}+\left(\dfrac{1}{3}\right)\sqrt{63}-\dfrac{29}{8}\right)-\left(\left(8\right)\sqrt{28}+\left(\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}\right)\right)-\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{70}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{47}{5}\right)\sqrt{63}+\left(-\dfrac{15}{2}\right)\sqrt{28}\right)\right)\\
&=&\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(\dfrac{76}{3}\right)\sqrt{7}+\left(1\right)\sqrt{7}-\dfrac{29}{8}\right)-\left(\left(16\right)\sqrt{7}+\left(\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}\right)\right)-\left(\left(\left(-10\right)\sqrt{7}+\left(\dfrac{140}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{141}{5}\right)\sqrt{7}+\left(-15\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{356}{3}\right)\sqrt{7}+\dfrac{29}{8}\right)-\left(\left(\dfrac{352}{15}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{356}{3}\right)\sqrt{7}+\dfrac{29}{8}+\left(-\dfrac{352}{15}\right)\sqrt{7}\\
&=&\left(-\dfrac{2132}{15}\right)\sqrt{7}+\dfrac{29}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-2\right)\sqrt{175}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{28}+\left(\dfrac{1}{3}\right)\sqrt{63}-\dfrac{29}{8}\right)-\left(\left(8\right)\sqrt{28}+\left(\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}\right)\right)\times\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{70}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{47}{5}\right)\sqrt{63}+\left(-\dfrac{15}{2}\right)\sqrt{28}\right)\right)\\
&=&\left(\left(\left(-10\right)\sqrt{7}\right)-\left(\left(\dfrac{76}{3}\right)\sqrt{7}+\left(1\right)\sqrt{7}-\dfrac{29}{8}\right)-\left(\left(16\right)\sqrt{7}+\left(\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}\right)\right)\times\left(\left(\left(-10\right)\sqrt{7}+\left(\dfrac{140}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{141}{5}\right)\sqrt{7}+\left(-15\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{356}{3}\right)\sqrt{7}+\dfrac{29}{8}\right)\left(\left(\dfrac{352}{15}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{125312}{45}\right)\sqrt{49}+\left(\dfrac{1276}{15}\right)\sqrt{7}\\
&=&-\dfrac{877184}{45}+\left(\dfrac{1276}{15}\right)\sqrt{7}\\
\end{eqnarray*}