L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{61}{2}\right)\sqrt{125}\right)-\dfrac{41}{2}-\left(\left(\dfrac{67}{9}\right)\sqrt{20}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{25}\right)+\left(-\dfrac{2}{3}\right)\sqrt{20}+\dfrac{41}{2}\) et \( Y=\left(\dfrac{15}{2}\right)\sqrt{45}\) . 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(\dfrac{61}{2}\right)\sqrt{125}\right)-\dfrac{41}{2}-\left(\left(\dfrac{67}{9}\right)\sqrt{20}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{25}\right)+\left(-\dfrac{2}{3}\right)\sqrt{20}+\dfrac{41}{2}\right)+\left(\left(\dfrac{15}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\dfrac{305}{2}\right)\sqrt{5}\right)-\dfrac{41}{2}-\left(\left(\dfrac{134}{9}\right)\sqrt{5}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{3}\right)\sqrt{5}\right)+\dfrac{70}{3}+\left(-\dfrac{4}{3}\right)\sqrt{5}+\dfrac{41}{2}\right)+\left(\left(\dfrac{45}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{305}{2}\right)\sqrt{5}\right)-\dfrac{41}{2}-\left(\left(\dfrac{134}{9}\right)\sqrt{5}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{3}\right)\sqrt{5}\right)+\dfrac{70}{3}+\left(-\dfrac{4}{3}\right)\sqrt{5}+\dfrac{41}{2}+\left(\dfrac{45}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{1255}{9}\right)\sqrt{5}+\dfrac{533}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{61}{2}\right)\sqrt{125}\right)-\dfrac{41}{2}-\left(\left(\dfrac{67}{9}\right)\sqrt{20}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{25}\right)+\left(-\dfrac{2}{3}\right)\sqrt{20}+\dfrac{41}{2}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\dfrac{305}{2}\right)\sqrt{5}\right)-\dfrac{41}{2}-\left(\left(\dfrac{134}{9}\right)\sqrt{5}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{3}\right)\sqrt{5}\right)+\dfrac{70}{3}+\left(-\dfrac{4}{3}\right)\sqrt{5}+\dfrac{41}{2}\right)-\left(\left(\dfrac{45}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{2105}{18}\right)\sqrt{5}+\dfrac{533}{15}\right)-\left(\left(\dfrac{45}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2105}{18}\right)\sqrt{5}+\dfrac{533}{15}+\left(-\dfrac{45}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{850}{9}\right)\sqrt{5}+\dfrac{533}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{61}{2}\right)\sqrt{125}\right)-\dfrac{41}{2}-\left(\left(\dfrac{67}{9}\right)\sqrt{20}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{9}\right)\sqrt{45}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{25}\right)+\left(-\dfrac{2}{3}\right)\sqrt{20}+\dfrac{41}{2}\right)\times\left(\left(\dfrac{15}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\dfrac{305}{2}\right)\sqrt{5}\right)-\dfrac{41}{2}-\left(\left(\dfrac{134}{9}\right)\sqrt{5}\right)+\dfrac{61}{5}-\left(\left(\dfrac{58}{3}\right)\sqrt{5}\right)+\dfrac{70}{3}+\left(-\dfrac{4}{3}\right)\sqrt{5}+\dfrac{41}{2}\right)\times\left(\left(\dfrac{45}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{2105}{18}\right)\sqrt{5}+\dfrac{533}{15}\right)\left(\left(\dfrac{45}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{10525}{4}\right)\sqrt{25}+\left(\dfrac{1599}{2}\right)\sqrt{5}\\
&=&\dfrac{52625}{4}+\left(\dfrac{1599}{2}\right)\sqrt{5}\\
\end{eqnarray*}