L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{18}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{8}\right)\sqrt{8}\right)\right)\) et \( Y=\left(\left(\left(\dfrac{26}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{50}\right)-\dfrac{64}{3}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{18}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{8}\right)\sqrt{8}\right)\right)\right)+\left(\left(\left(\left(\dfrac{26}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{50}\right)-\dfrac{64}{3}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{4}\right)\sqrt{2}\right)\right)\right)+\left(\left(\left(\left(26\right)\sqrt{2}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)-\dfrac{64}{3}\right)-\dfrac{35}{2}\right)\\
&=&\left(\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{4}\right)\sqrt{2}\right)\right)+\left(\left(\left(26\right)\sqrt{2}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)-\dfrac{64}{3}\right)-\dfrac{35}{2}\\
&=&\left(\dfrac{105}{4}\right)\sqrt{2}-\dfrac{221}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{18}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{8}\right)\sqrt{8}\right)\right)\right)-\left(\left(\left(\left(\dfrac{26}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{50}\right)-\dfrac{64}{3}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{4}\right)\sqrt{2}\right)\right)\right)-\left(\left(\left(\left(26\right)\sqrt{2}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)-\dfrac{64}{3}\right)-\dfrac{35}{2}\right)\\
&=&\left(\left(-\dfrac{89}{4}\right)\sqrt{2}+2\right)-\left(\left(\dfrac{97}{2}\right)\sqrt{2}-\dfrac{233}{6}\right)\\
&=&\left(-\dfrac{89}{4}\right)\sqrt{2}+2+\left(-\dfrac{97}{2}\right)\sqrt{2}+\dfrac{233}{6}\\
&=&\left(-\dfrac{283}{4}\right)\sqrt{2}+\dfrac{245}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-5\right)\sqrt{50}\right)-\left(\left(-6\right)\sqrt{18}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{8}\right)\sqrt{8}\right)\right)\right)\times\left(\left(\left(\left(\dfrac{26}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{50}\right)-\dfrac{64}{3}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(\left(-25\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)-\left(-2-\left(\left(-\dfrac{61}{4}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\left(\left(26\right)\sqrt{2}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)-\dfrac{64}{3}\right)-\dfrac{35}{2}\right)\\
&=&\left(\left(-\dfrac{89}{4}\right)\sqrt{2}+2\right)\left(\left(\dfrac{97}{2}\right)\sqrt{2}-\dfrac{233}{6}\right)\\
&=&\left(-\dfrac{8633}{8}\right)\sqrt{4}+\left(\dfrac{23065}{24}\right)\sqrt{2}-\dfrac{233}{3}\\
&=&-\dfrac{26831}{12}+\left(\dfrac{23065}{24}\right)\sqrt{2}\\
\end{eqnarray*}