L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{75}{8}+\left(\left(\dfrac{19}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{6}\right)\sqrt{18}\right)+\left(\dfrac{48}{7}\right)\sqrt{8}+\left(-\dfrac{2}{3}\right)\sqrt{18}+\left(-\dfrac{71}{9}\right)\sqrt{18}\) et \( Y=\left(\dfrac{35}{4}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{75}{8}+\left(\left(\dfrac{19}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{6}\right)\sqrt{18}\right)+\left(\dfrac{48}{7}\right)\sqrt{8}+\left(-\dfrac{2}{3}\right)\sqrt{18}+\left(-\dfrac{71}{9}\right)\sqrt{18}\right)+\left(\left(\dfrac{35}{4}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{75}{8}+\left(\left(\dfrac{38}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{53}{2}\right)\sqrt{2}\right)+\left(\dfrac{96}{7}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+\left(-\dfrac{71}{3}\right)\sqrt{2}\right)+\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)\\
&=&\dfrac{75}{8}+\left(\left(\dfrac{38}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{53}{2}\right)\sqrt{2}\right)+\left(\dfrac{96}{7}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+\left(-\dfrac{71}{3}\right)\sqrt{2}+\left(\dfrac{105}{4}\right)\sqrt{2}\\
&=&\dfrac{75}{8}+\left(\dfrac{16657}{1260}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{75}{8}+\left(\left(\dfrac{19}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{6}\right)\sqrt{18}\right)+\left(\dfrac{48}{7}\right)\sqrt{8}+\left(-\dfrac{2}{3}\right)\sqrt{18}+\left(-\dfrac{71}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{75}{8}+\left(\left(\dfrac{38}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{53}{2}\right)\sqrt{2}\right)+\left(\dfrac{96}{7}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+\left(-\dfrac{71}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{75}{8}+\left(-\dfrac{8209}{630}\right)\sqrt{2}\right)-\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)\\
&=&\dfrac{75}{8}+\left(-\dfrac{8209}{630}\right)\sqrt{2}+\left(-\dfrac{105}{4}\right)\sqrt{2}\\
&=&\dfrac{75}{8}+\left(-\dfrac{49493}{1260}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{75}{8}+\left(\left(\dfrac{19}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{53}{6}\right)\sqrt{18}\right)+\left(\dfrac{48}{7}\right)\sqrt{8}+\left(-\dfrac{2}{3}\right)\sqrt{18}+\left(-\dfrac{71}{9}\right)\sqrt{18}\right)\times\left(\left(\dfrac{35}{4}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{75}{8}+\left(\left(\dfrac{38}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{106}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{53}{2}\right)\sqrt{2}\right)+\left(\dfrac{96}{7}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+\left(-\dfrac{71}{3}\right)\sqrt{2}\right)\times\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{75}{8}+\left(-\dfrac{8209}{630}\right)\sqrt{2}\right)\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{7875}{32}\right)\sqrt{2}+\left(-\dfrac{8209}{24}\right)\sqrt{4}\\
&=&\left(\dfrac{7875}{32}\right)\sqrt{2}-\dfrac{8209}{12}\\
\end{eqnarray*}