L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-4\right)\sqrt{25}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{45}\) et \( Y=\left(\left(\dfrac{32}{7}\right)\sqrt{125}-\dfrac{1}{2}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{45}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-4\right)\sqrt{25}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{45}\right)+\left(\left(\left(\dfrac{32}{7}\right)\sqrt{125}-\dfrac{1}{2}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{45}\right)\right)\\
&=&\left(-\dfrac{260}{3}+\left(35\right)\sqrt{5}-20+\dfrac{50}{3}+\left(35\right)\sqrt{5}+\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(23\right)\sqrt{5}\right)+\left(\left(\left(\dfrac{160}{7}\right)\sqrt{5}-\dfrac{1}{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{5}\right)\right)\\
&=&-\dfrac{260}{3}+\left(35\right)\sqrt{5}-20+\dfrac{50}{3}+\left(35\right)\sqrt{5}+\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(23\right)\sqrt{5}+\left(\left(\dfrac{160}{7}\right)\sqrt{5}-\dfrac{1}{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{5}\right)\\
&=&-\dfrac{181}{2}+\left(\dfrac{1599}{28}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-4\right)\sqrt{25}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{32}{7}\right)\sqrt{125}-\dfrac{1}{2}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{45}\right)\right)\\
&=&\left(-\dfrac{260}{3}+\left(35\right)\sqrt{5}-20+\dfrac{50}{3}+\left(35\right)\sqrt{5}+\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(23\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{160}{7}\right)\sqrt{5}-\dfrac{1}{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(-90+\left(\dfrac{91}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{325}{28}\right)\sqrt{5}-\dfrac{1}{2}\right)\\
&=&-90+\left(\dfrac{91}{2}\right)\sqrt{5}+\left(-\dfrac{325}{28}\right)\sqrt{5}+\dfrac{1}{2}\\
&=&-\dfrac{179}{2}+\left(\dfrac{949}{28}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{52}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-4\right)\sqrt{25}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(7\right)\sqrt{125}+\left(-\dfrac{19}{2}\right)\sqrt{125}+\left(\dfrac{23}{3}\right)\sqrt{45}\right)\times\left(\left(\left(\dfrac{32}{7}\right)\sqrt{125}-\dfrac{1}{2}\right)-\left(\left(\dfrac{15}{4}\right)\sqrt{45}\right)\right)\\
&=&\left(-\dfrac{260}{3}+\left(35\right)\sqrt{5}-20+\dfrac{50}{3}+\left(35\right)\sqrt{5}+\left(-\dfrac{95}{2}\right)\sqrt{5}+\left(23\right)\sqrt{5}\right)\times\left(\left(\left(\dfrac{160}{7}\right)\sqrt{5}-\dfrac{1}{2}\right)-\left(\left(\dfrac{45}{4}\right)\sqrt{5}\right)\right)\\
&=&\left(-90+\left(\dfrac{91}{2}\right)\sqrt{5}\right)\left(\left(\dfrac{325}{28}\right)\sqrt{5}-\dfrac{1}{2}\right)\\
&=&\left(-\dfrac{29887}{28}\right)\sqrt{5}+45+\left(\dfrac{4225}{8}\right)\sqrt{25}\\
&=&\left(-\dfrac{29887}{28}\right)\sqrt{5}+\dfrac{21485}{8}\\
\end{eqnarray*}