L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{39}{5}\right)\sqrt{18}\) et \( Y=\left(\left(-7\right)\sqrt{4}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)+\left(-6\right)\sqrt{18}+\left(\left(-7\right)\sqrt{50}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{70}{9}\right)\sqrt{50}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{18}\right)+\left(\dfrac{64}{9}\right)\sqrt{18}\) . 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{39}{5}\right)\sqrt{18}\right)+\left(\left(\left(-7\right)\sqrt{4}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)+\left(-6\right)\sqrt{18}+\left(\left(-7\right)\sqrt{50}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{70}{9}\right)\sqrt{50}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{18}\right)+\left(\dfrac{64}{9}\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{117}{5}\right)\sqrt{2}\right)+\left(-14-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\left(-35\right)\sqrt{2}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{350}{9}\right)\sqrt{2}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)+\left(\dfrac{64}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{117}{5}\right)\sqrt{2}-14-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\left(-35\right)\sqrt{2}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{350}{9}\right)\sqrt{2}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)+\left(\dfrac{64}{3}\right)\sqrt{2}\\
&=&\left(-\dfrac{2428}{45}\right)\sqrt{2}-\dfrac{20}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{39}{5}\right)\sqrt{18}\right)-\left(\left(\left(-7\right)\sqrt{4}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)+\left(-6\right)\sqrt{18}+\left(\left(-7\right)\sqrt{50}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{70}{9}\right)\sqrt{50}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{18}\right)+\left(\dfrac{64}{9}\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{117}{5}\right)\sqrt{2}\right)-\left(-14-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\left(-35\right)\sqrt{2}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{350}{9}\right)\sqrt{2}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)+\left(\dfrac{64}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{117}{5}\right)\sqrt{2}\right)-\left(-\dfrac{20}{3}+\left(-\dfrac{275}{9}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{117}{5}\right)\sqrt{2}+\dfrac{20}{3}+\left(\dfrac{275}{9}\right)\sqrt{2}\\
&=&\left(\dfrac{322}{45}\right)\sqrt{2}+\dfrac{20}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{39}{5}\right)\sqrt{18}\right)\times\left(\left(\left(-7\right)\sqrt{4}\right)-\left(\left(\dfrac{74}{9}\right)\sqrt{8}\right)+\left(-6\right)\sqrt{18}+\left(\left(-7\right)\sqrt{50}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{70}{9}\right)\sqrt{50}\right)-\left(\left(\dfrac{64}{9}\right)\sqrt{18}\right)+\left(\dfrac{64}{9}\right)\sqrt{18}\right)\\
&=&\left(\left(-\dfrac{117}{5}\right)\sqrt{2}\right)\times\left(-14-\left(\left(\dfrac{148}{9}\right)\sqrt{2}\right)+\left(-18\right)\sqrt{2}+\left(\left(-35\right)\sqrt{2}\right)+\dfrac{22}{3}-\left(\left(-\dfrac{350}{9}\right)\sqrt{2}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)+\left(\dfrac{64}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{117}{5}\right)\sqrt{2}\right)\left(-\dfrac{20}{3}+\left(-\dfrac{275}{9}\right)\sqrt{2}\right)\\
&=&\left(156\right)\sqrt{2}+\left(715\right)\sqrt{4}\\
&=&\left(156\right)\sqrt{2}+1430\\
\end{eqnarray*}