L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{1}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{25}\right)-\left(\left(\left(-8\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{5}\right)\sqrt{25}\right)\right)\) et \( Y=\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{58}{3}\right)\sqrt{20}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-7\right)\sqrt{125}\right)+\dfrac{11}{2}\right)\) . 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(-\dfrac{1}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{25}\right)-\left(\left(\left(-8\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{5}\right)\sqrt{25}\right)\right)\right)+\left(\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{58}{3}\right)\sqrt{20}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-7\right)\sqrt{125}\right)+\dfrac{11}{2}\right)\right)\\
&=&\left(-\dfrac{5}{2}-73-\left(\left(\left(-40\right)\sqrt{5}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}\right)-49\right)\right)+\left(-45-\left(\left(-\dfrac{116}{3}\right)\sqrt{5}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-35\right)\sqrt{5}\right)+\dfrac{11}{2}\right)\right)\\
&=&-\dfrac{5}{2}-73-\left(\left(\left(-40\right)\sqrt{5}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}\right)-49\right)-45-\left(\left(-\dfrac{116}{3}\right)\sqrt{5}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-35\right)\sqrt{5}\right)+\dfrac{11}{2}\right)\\
&=&-\dfrac{159}{2}+\left(\dfrac{229}{6}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{25}\right)-\left(\left(\left(-8\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{5}\right)\sqrt{25}\right)\right)\right)-\left(\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{58}{3}\right)\sqrt{20}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-7\right)\sqrt{125}\right)+\dfrac{11}{2}\right)\right)\\
&=&\left(-\dfrac{5}{2}-73-\left(\left(\left(-40\right)\sqrt{5}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}\right)-49\right)\right)-\left(-45-\left(\left(-\dfrac{116}{3}\right)\sqrt{5}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-35\right)\sqrt{5}\right)+\dfrac{11}{2}\right)\right)\\
&=&\left(-\dfrac{53}{2}+\left(\dfrac{69}{2}\right)\sqrt{5}\right)-\left(-53+\left(\dfrac{11}{3}\right)\sqrt{5}\right)\\
&=&-\dfrac{53}{2}+\left(\dfrac{69}{2}\right)\sqrt{5}+53+\left(-\dfrac{11}{3}\right)\sqrt{5}\\
&=&\dfrac{53}{2}+\left(\dfrac{185}{6}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{73}{5}\right)\sqrt{25}\right)-\left(\left(\left(-8\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{49}{5}\right)\sqrt{25}\right)\right)\right)\times\left(\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{58}{3}\right)\sqrt{20}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-7\right)\sqrt{125}\right)+\dfrac{11}{2}\right)\right)\\
&=&\left(-\dfrac{5}{2}-73-\left(\left(\left(-40\right)\sqrt{5}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{5}\right)-49\right)\right)\times\left(-45-\left(\left(-\dfrac{116}{3}\right)\sqrt{5}\right)-\left(-\dfrac{11}{2}+8-\left(\left(-35\right)\sqrt{5}\right)+\dfrac{11}{2}\right)\right)\\
&=&\left(-\dfrac{53}{2}+\left(\dfrac{69}{2}\right)\sqrt{5}\right)\left(-53+\left(\dfrac{11}{3}\right)\sqrt{5}\right)\\
&=&\dfrac{2809}{2}+\left(-\dfrac{5777}{3}\right)\sqrt{5}+\left(\dfrac{253}{2}\right)\sqrt{25}\\
&=&2037+\left(-\dfrac{5777}{3}\right)\sqrt{5}\\
\end{eqnarray*}