L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(\dfrac{15}{8}\right)\sqrt{8}+\left(4\right)\sqrt{18}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-8\right)\sqrt{8}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{8}+\left(-\dfrac{37}{5}\right)\sqrt{8}+\left(-\dfrac{67}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{79}{8}\right)\sqrt{4}\right)\) et \( Y=\left(\dfrac{47}{9}\right)\sqrt{8}\) . 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{75}{4}\right)\sqrt{8}+\left(\dfrac{15}{8}\right)\sqrt{8}+\left(4\right)\sqrt{18}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-8\right)\sqrt{8}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{8}+\left(-\dfrac{37}{5}\right)\sqrt{8}+\left(-\dfrac{67}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{79}{8}\right)\sqrt{4}\right)\right)+\left(\left(\dfrac{47}{9}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(-\dfrac{75}{2}\right)\sqrt{2}+\left(\dfrac{15}{4}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-16\right)\sqrt{2}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{2}+\left(-\dfrac{74}{5}\right)\sqrt{2}+\left(-\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{79}{4}\right)+\left(\left(\dfrac{94}{9}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{75}{2}\right)\sqrt{2}+\left(\dfrac{15}{4}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-16\right)\sqrt{2}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{2}+\left(-\dfrac{74}{5}\right)\sqrt{2}+\left(-\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{79}{4}+\left(\dfrac{94}{9}\right)\sqrt{2}\\
&=&\left(\dfrac{2843}{60}\right)\sqrt{2}+\dfrac{377}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(\dfrac{15}{8}\right)\sqrt{8}+\left(4\right)\sqrt{18}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-8\right)\sqrt{8}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{8}+\left(-\dfrac{37}{5}\right)\sqrt{8}+\left(-\dfrac{67}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{79}{8}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{47}{9}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(-\dfrac{75}{2}\right)\sqrt{2}+\left(\dfrac{15}{4}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-16\right)\sqrt{2}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{2}+\left(-\dfrac{74}{5}\right)\sqrt{2}+\left(-\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{79}{4}\right)-\left(\left(\dfrac{94}{9}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{6649}{180}\right)\sqrt{2}+\dfrac{377}{20}\right)-\left(\left(\dfrac{94}{9}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{6649}{180}\right)\sqrt{2}+\dfrac{377}{20}+\left(-\dfrac{94}{9}\right)\sqrt{2}\\
&=&\left(\dfrac{4769}{180}\right)\sqrt{2}+\dfrac{377}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{75}{4}\right)\sqrt{8}+\left(\dfrac{15}{8}\right)\sqrt{8}+\left(4\right)\sqrt{18}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-8\right)\sqrt{8}\right)-\left(\left(\dfrac{14}{3}\right)\sqrt{8}+\left(-\dfrac{37}{5}\right)\sqrt{8}+\left(-\dfrac{67}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{79}{8}\right)\sqrt{4}\right)\right)\times\left(\left(\dfrac{47}{9}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(-\dfrac{75}{2}\right)\sqrt{2}+\left(\dfrac{15}{4}\right)\sqrt{2}+\left(12\right)\sqrt{2}+\dfrac{8}{5}\right)-\left(\dfrac{5}{2}+\left(-16\right)\sqrt{2}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{2}+\left(-\dfrac{74}{5}\right)\sqrt{2}+\left(-\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{79}{4}\right)\times\left(\left(\dfrac{94}{9}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{6649}{180}\right)\sqrt{2}+\dfrac{377}{20}\right)\left(\left(\dfrac{94}{9}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{312503}{810}\right)\sqrt{4}+\left(\dfrac{17719}{90}\right)\sqrt{2}\\
&=&\dfrac{312503}{405}+\left(\dfrac{17719}{90}\right)\sqrt{2}\\
\end{eqnarray*}