L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{15}{4}\right)\sqrt{125}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(4\right)\sqrt{125}+0\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{20}\right)\) et \( Y=\left(-\dfrac{79}{8}\right)\sqrt{125}\) . 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{15}{4}\right)\sqrt{125}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(4\right)\sqrt{125}+0\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{20}\right)\right)+\left(\left(-\dfrac{79}{8}\right)\sqrt{125}\right)\\
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(20\right)\sqrt{5}+0\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{5}\right)\right)+\left(\left(-\dfrac{395}{8}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{75}{4}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(20\right)\sqrt{5}+0\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{5}\right)+\left(-\dfrac{395}{8}\right)\sqrt{5}\\
&=&\left(-\dfrac{919}{24}\right)\sqrt{5}+0\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{125}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(4\right)\sqrt{125}+0\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{79}{8}\right)\sqrt{125}\right)\\
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(20\right)\sqrt{5}+0\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{395}{8}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{133}{12}\right)\sqrt{5}+0\right)-\left(\left(-\dfrac{395}{8}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{133}{12}\right)\sqrt{5}+0+\left(\dfrac{395}{8}\right)\sqrt{5}\\
&=&\left(\dfrac{1451}{24}\right)\sqrt{5}+0\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{125}+\left(\dfrac{3}{2}\right)\sqrt{125}+\left(4\right)\sqrt{125}+0\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{20}\right)\right)\times\left(\left(-\dfrac{79}{8}\right)\sqrt{125}\right)\\
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{5}+\left(\dfrac{15}{2}\right)\sqrt{5}+\left(20\right)\sqrt{5}+0\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{5}\right)\right)\times\left(\left(-\dfrac{395}{8}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{133}{12}\right)\sqrt{5}+0\right)\left(\left(-\dfrac{395}{8}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{52535}{96}\right)\sqrt{25}+\left(0\right)\sqrt{5}\\
&=&-\dfrac{262675}{96}+\left(0\right)\sqrt{5}\\
\end{eqnarray*}