L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{41}{8}\right)\sqrt{28}-\dfrac{25}{8}+\left(-\dfrac{78}{5}\right)\sqrt{28}+\left(-6\right)\sqrt{175}+\left(0\right)\sqrt{49}-\dfrac{65}{6}+\left(3\right)\sqrt{175}\) et \( Y=\left(\dfrac{11}{2}\right)\sqrt{175}\) . 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{41}{8}\right)\sqrt{28}-\dfrac{25}{8}+\left(-\dfrac{78}{5}\right)\sqrt{28}+\left(-6\right)\sqrt{175}+\left(0\right)\sqrt{49}-\dfrac{65}{6}+\left(3\right)\sqrt{175}\right)+\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)\\
&=&\left(\left(\dfrac{41}{4}\right)\sqrt{7}-\dfrac{25}{8}+\left(-\dfrac{156}{5}\right)\sqrt{7}+\left(-30\right)\sqrt{7}+0-\dfrac{65}{6}+\left(15\right)\sqrt{7}\right)+\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{41}{4}\right)\sqrt{7}-\dfrac{25}{8}+\left(-\dfrac{156}{5}\right)\sqrt{7}+\left(-30\right)\sqrt{7}+0-\dfrac{65}{6}+\left(15\right)\sqrt{7}+\left(\dfrac{55}{2}\right)\sqrt{7}\\
&=&\left(-\dfrac{169}{20}\right)\sqrt{7}-\dfrac{335}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{41}{8}\right)\sqrt{28}-\dfrac{25}{8}+\left(-\dfrac{78}{5}\right)\sqrt{28}+\left(-6\right)\sqrt{175}+\left(0\right)\sqrt{49}-\dfrac{65}{6}+\left(3\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)\\
&=&\left(\left(\dfrac{41}{4}\right)\sqrt{7}-\dfrac{25}{8}+\left(-\dfrac{156}{5}\right)\sqrt{7}+\left(-30\right)\sqrt{7}+0-\dfrac{65}{6}+\left(15\right)\sqrt{7}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)\\
&=&\left(\left(-\dfrac{719}{20}\right)\sqrt{7}-\dfrac{335}{24}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{719}{20}\right)\sqrt{7}-\dfrac{335}{24}+\left(-\dfrac{55}{2}\right)\sqrt{7}\\
&=&\left(-\dfrac{1269}{20}\right)\sqrt{7}-\dfrac{335}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{41}{8}\right)\sqrt{28}-\dfrac{25}{8}+\left(-\dfrac{78}{5}\right)\sqrt{28}+\left(-6\right)\sqrt{175}+\left(0\right)\sqrt{49}-\dfrac{65}{6}+\left(3\right)\sqrt{175}\right)\times\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)\\
&=&\left(\left(\dfrac{41}{4}\right)\sqrt{7}-\dfrac{25}{8}+\left(-\dfrac{156}{5}\right)\sqrt{7}+\left(-30\right)\sqrt{7}+0-\dfrac{65}{6}+\left(15\right)\sqrt{7}\right)\times\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)\\
&=&\left(\left(-\dfrac{719}{20}\right)\sqrt{7}-\dfrac{335}{24}\right)\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{7909}{8}\right)\sqrt{49}+\left(-\dfrac{18425}{48}\right)\sqrt{7}\\
&=&-\dfrac{55363}{8}+\left(-\dfrac{18425}{48}\right)\sqrt{7}\\
\end{eqnarray*}