L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{4}\right)+\left(-\dfrac{48}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}+\left(1\right)\sqrt{8}\) et \( Y=\left(-\dfrac{41}{5}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{4}\right)+\left(-\dfrac{48}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}+\left(1\right)\sqrt{8}\right)+\left(\left(-\dfrac{41}{5}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)-\dfrac{26}{9}+\left(-\dfrac{96}{5}\right)\sqrt{2}+0+\left(2\right)\sqrt{2}\right)+\left(\left(-\dfrac{123}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)-\dfrac{26}{9}+\left(-\dfrac{96}{5}\right)\sqrt{2}+0+\left(2\right)\sqrt{2}+\left(-\dfrac{123}{5}\right)\sqrt{2}\\
&=&\left(-\dfrac{143}{5}\right)\sqrt{2}-\dfrac{26}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{4}\right)+\left(-\dfrac{48}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}+\left(1\right)\sqrt{8}\right)-\left(\left(-\dfrac{41}{5}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)-\dfrac{26}{9}+\left(-\dfrac{96}{5}\right)\sqrt{2}+0+\left(2\right)\sqrt{2}\right)-\left(\left(-\dfrac{123}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-4\right)\sqrt{2}-\dfrac{26}{9}\right)-\left(\left(-\dfrac{123}{5}\right)\sqrt{2}\right)\\
&=&\left(-4\right)\sqrt{2}-\dfrac{26}{9}+\left(\dfrac{123}{5}\right)\sqrt{2}\\
&=&\left(\dfrac{103}{5}\right)\sqrt{2}-\dfrac{26}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{33}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{4}\right)+\left(-\dfrac{48}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}+\left(1\right)\sqrt{8}\right)\times\left(\left(-\dfrac{41}{5}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(\dfrac{66}{5}\right)\sqrt{2}\right)-\dfrac{26}{9}+\left(-\dfrac{96}{5}\right)\sqrt{2}+0+\left(2\right)\sqrt{2}\right)\times\left(\left(-\dfrac{123}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-4\right)\sqrt{2}-\dfrac{26}{9}\right)\left(\left(-\dfrac{123}{5}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{492}{5}\right)\sqrt{4}+\left(\dfrac{1066}{15}\right)\sqrt{2}\\
&=&\dfrac{984}{5}+\left(\dfrac{1066}{15}\right)\sqrt{2}\\
\end{eqnarray*}