L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{1}{2}\right)\sqrt{8}+\left(-3\right)\sqrt{18}+\left(-\dfrac{39}{4}\right)\sqrt{4}+\left(\dfrac{56}{5}\right)\sqrt{18}+\left(\dfrac{55}{3}\right)\sqrt{4}+\left(0\right)\sqrt{50}\) et \( Y=\left(\dfrac{11}{7}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{1}{2}\right)\sqrt{8}+\left(-3\right)\sqrt{18}+\left(-\dfrac{39}{4}\right)\sqrt{4}+\left(\dfrac{56}{5}\right)\sqrt{18}+\left(\dfrac{55}{3}\right)\sqrt{4}+\left(0\right)\sqrt{50}\right)+\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)\\
&=&\left(\left(-1\right)\sqrt{2}+\left(-9\right)\sqrt{2}-\dfrac{39}{2}+\left(\dfrac{168}{5}\right)\sqrt{2}+\dfrac{110}{3}+\left(0\right)\sqrt{2}\right)+\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)\\
&=&\left(-1\right)\sqrt{2}+\left(-9\right)\sqrt{2}-\dfrac{39}{2}+\left(\dfrac{168}{5}\right)\sqrt{2}+\dfrac{110}{3}+\left(0\right)\sqrt{2}+\left(\dfrac{33}{7}\right)\sqrt{2}\\
&=&\left(\dfrac{991}{35}\right)\sqrt{2}+\dfrac{103}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{1}{2}\right)\sqrt{8}+\left(-3\right)\sqrt{18}+\left(-\dfrac{39}{4}\right)\sqrt{4}+\left(\dfrac{56}{5}\right)\sqrt{18}+\left(\dfrac{55}{3}\right)\sqrt{4}+\left(0\right)\sqrt{50}\right)-\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)\\
&=&\left(\left(-1\right)\sqrt{2}+\left(-9\right)\sqrt{2}-\dfrac{39}{2}+\left(\dfrac{168}{5}\right)\sqrt{2}+\dfrac{110}{3}+\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{118}{5}\right)\sqrt{2}+\dfrac{103}{6}\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{118}{5}\right)\sqrt{2}+\dfrac{103}{6}+\left(-\dfrac{33}{7}\right)\sqrt{2}\\
&=&\left(\dfrac{661}{35}\right)\sqrt{2}+\dfrac{103}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{1}{2}\right)\sqrt{8}+\left(-3\right)\sqrt{18}+\left(-\dfrac{39}{4}\right)\sqrt{4}+\left(\dfrac{56}{5}\right)\sqrt{18}+\left(\dfrac{55}{3}\right)\sqrt{4}+\left(0\right)\sqrt{50}\right)\times\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)\\
&=&\left(\left(-1\right)\sqrt{2}+\left(-9\right)\sqrt{2}-\dfrac{39}{2}+\left(\dfrac{168}{5}\right)\sqrt{2}+\dfrac{110}{3}+\left(0\right)\sqrt{2}\right)\times\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{118}{5}\right)\sqrt{2}+\dfrac{103}{6}\right)\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{3894}{35}\right)\sqrt{4}+\left(\dfrac{1133}{14}\right)\sqrt{2}\\
&=&\dfrac{7788}{35}+\left(\dfrac{1133}{14}\right)\sqrt{2}\\
\end{eqnarray*}