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 \( X=\left(\dfrac{65}{6}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{27}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{13}{3}-\left(\left(\dfrac{29}{7}\right)\sqrt{25}\right)\) et \( Y=\dfrac{55}{3}\) . 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{65}{6}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{27}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{13}{3}-\left(\left(\dfrac{29}{7}\right)\sqrt{25}\right)\right)+\left(\dfrac{55}{3}\right)\\
&=&\left(\left(\dfrac{325}{6}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{13}{3}-\dfrac{145}{7}\right)+\left(\dfrac{55}{3}\right)\\
&=&\left(\dfrac{325}{6}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{13}{3}-\dfrac{145}{7}+\dfrac{55}{3}\\
&=&\left(\dfrac{3406}{105}\right)\sqrt{5}+\dfrac{95}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{65}{6}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{27}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{13}{3}-\left(\left(\dfrac{29}{7}\right)\sqrt{25}\right)\right)-\left(\dfrac{55}{3}\right)\\
&=&\left(\left(\dfrac{325}{6}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{13}{3}-\dfrac{145}{7}\right)-\left(\dfrac{55}{3}\right)\\
&=&\left(\left(\dfrac{3406}{105}\right)\sqrt{5}-\dfrac{485}{42}\right)-\left(\dfrac{55}{3}\right)\\
&=&\left(\dfrac{3406}{105}\right)\sqrt{5}-\dfrac{485}{42}+-\dfrac{55}{3}\\
&=&\left(\dfrac{3406}{105}\right)\sqrt{5}-\dfrac{1255}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{65}{6}\right)\sqrt{125}+\left(\dfrac{15}{2}\right)\sqrt{45}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{27}{5}\right)\sqrt{20}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{45}\right)-\dfrac{13}{3}-\left(\left(\dfrac{29}{7}\right)\sqrt{25}\right)\right)\times\left(\dfrac{55}{3}\right)\\
&=&\left(\left(\dfrac{325}{6}\right)\sqrt{5}+\left(\dfrac{45}{2}\right)\sqrt{5}+6+\dfrac{27}{2}-6+\left(\left(-\dfrac{54}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{234}{7}\right)\sqrt{5}\right)-\dfrac{13}{3}-\dfrac{145}{7}\right)\times\left(\dfrac{55}{3}\right)\\
&=&\left(\left(\dfrac{3406}{105}\right)\sqrt{5}-\dfrac{485}{42}\right)\left(\dfrac{55}{3}\right)\\
&=&\left(\dfrac{37466}{63}\right)\sqrt{5}-\dfrac{26675}{126}\\
&=&\left(\dfrac{37466}{63}\right)\sqrt{5}-\dfrac{26675}{126}\\
\end{eqnarray*}