L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(3\right)\sqrt{8}+\left(\left(-\dfrac{5}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{8}\right)\) et \( Y=\left(\dfrac{3}{2}\right)\sqrt{8}+\left(-\dfrac{55}{9}\right)\sqrt{18}+\left(\dfrac{39}{4}\right)\sqrt{50}+\left(\dfrac{33}{4}\right)\sqrt{4}+\left(\dfrac{74}{3}\right)\sqrt{4}+\left(-\dfrac{61}{2}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(3\right)\sqrt{8}+\left(\left(-\dfrac{5}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{8}\right)\right)+\left(\left(\dfrac{3}{2}\right)\sqrt{8}+\left(-\dfrac{55}{9}\right)\sqrt{18}+\left(\dfrac{39}{4}\right)\sqrt{50}+\left(\dfrac{33}{4}\right)\sqrt{4}+\left(\dfrac{74}{3}\right)\sqrt{4}+\left(-\dfrac{61}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(6\right)\sqrt{2}+\left(\left(-\dfrac{15}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{2}\right)\right)+\left(\left(3\right)\sqrt{2}+\left(-\dfrac{55}{3}\right)\sqrt{2}+\left(\dfrac{195}{4}\right)\sqrt{2}+\dfrac{33}{2}+\dfrac{148}{3}+\left(-\dfrac{305}{2}\right)\sqrt{2}\right)\\
&=&\left(6\right)\sqrt{2}+\left(\left(-\dfrac{15}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{2}\right)+\left(3\right)\sqrt{2}+\left(-\dfrac{55}{3}\right)\sqrt{2}+\left(\dfrac{195}{4}\right)\sqrt{2}+\dfrac{33}{2}+\dfrac{148}{3}+\left(-\dfrac{305}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{319}{3}\right)\sqrt{2}+\dfrac{395}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(3\right)\sqrt{8}+\left(\left(-\dfrac{5}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{8}\right)\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{8}+\left(-\dfrac{55}{9}\right)\sqrt{18}+\left(\dfrac{39}{4}\right)\sqrt{50}+\left(\dfrac{33}{4}\right)\sqrt{4}+\left(\dfrac{74}{3}\right)\sqrt{4}+\left(-\dfrac{61}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(6\right)\sqrt{2}+\left(\left(-\dfrac{15}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{2}\right)\right)-\left(\left(3\right)\sqrt{2}+\left(-\dfrac{55}{3}\right)\sqrt{2}+\left(\dfrac{195}{4}\right)\sqrt{2}+\dfrac{33}{2}+\dfrac{148}{3}+\left(-\dfrac{305}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{51}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{1429}{12}\right)\sqrt{2}+\dfrac{395}{6}\right)\\
&=&\left(\dfrac{51}{4}\right)\sqrt{2}+\left(\dfrac{1429}{12}\right)\sqrt{2}-\dfrac{395}{6}\\
&=&\left(\dfrac{791}{6}\right)\sqrt{2}-\dfrac{395}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(3\right)\sqrt{8}+\left(\left(-\dfrac{5}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{8}\right)\right)\times\left(\left(\dfrac{3}{2}\right)\sqrt{8}+\left(-\dfrac{55}{9}\right)\sqrt{18}+\left(\dfrac{39}{4}\right)\sqrt{50}+\left(\dfrac{33}{4}\right)\sqrt{4}+\left(\dfrac{74}{3}\right)\sqrt{4}+\left(-\dfrac{61}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(6\right)\sqrt{2}+\left(\left(-\dfrac{15}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{2}\right)\sqrt{2}\right)\right)\times\left(\left(3\right)\sqrt{2}+\left(-\dfrac{55}{3}\right)\sqrt{2}+\left(\dfrac{195}{4}\right)\sqrt{2}+\dfrac{33}{2}+\dfrac{148}{3}+\left(-\dfrac{305}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{51}{4}\right)\sqrt{2}\right)\left(\left(-\dfrac{1429}{12}\right)\sqrt{2}+\dfrac{395}{6}\right)\\
&=&\left(-\dfrac{24293}{16}\right)\sqrt{4}+\left(\dfrac{6715}{8}\right)\sqrt{2}\\
&=&-\dfrac{24293}{8}+\left(\dfrac{6715}{8}\right)\sqrt{2}\\
\end{eqnarray*}