L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{15}{7}\) et \( Y=\left(\left(\dfrac{65}{8}\right)\sqrt{49}+\left(-\dfrac{33}{4}\right)\sqrt{49}+\left(-\dfrac{34}{5}\right)\sqrt{63}\right)-\left(9-\left(\left(3\right)\sqrt{175}\right)\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{175}+\left(\dfrac{79}{8}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}-\dfrac{47}{9}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{15}{7}\right)+\left(\left(\left(\dfrac{65}{8}\right)\sqrt{49}+\left(-\dfrac{33}{4}\right)\sqrt{49}+\left(-\dfrac{34}{5}\right)\sqrt{63}\right)-\left(9-\left(\left(3\right)\sqrt{175}\right)\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{175}+\left(\dfrac{79}{8}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}-\dfrac{47}{9}\right)\right)\\
&=&\left(\dfrac{15}{7}\right)+\left(\left(\dfrac{455}{8}-\dfrac{231}{4}+\left(-\dfrac{102}{5}\right)\sqrt{7}\right)-\left(9-\left(\left(15\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{205}{2}\right)\sqrt{7}+\left(\dfrac{395}{8}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{180}{7}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}-\dfrac{47}{9}\right)\right)\\
&=&\dfrac{15}{7}+\left(\dfrac{455}{8}-\dfrac{231}{4}+\left(-\dfrac{102}{5}\right)\sqrt{7}\right)-\left(9-\left(\left(15\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{205}{2}\right)\sqrt{7}+\left(\dfrac{395}{8}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{180}{7}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}-\dfrac{47}{9}\right)\\
&=&-\dfrac{1265}{504}+\left(-\dfrac{9891}{40}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{15}{7}\right)-\left(\left(\left(\dfrac{65}{8}\right)\sqrt{49}+\left(-\dfrac{33}{4}\right)\sqrt{49}+\left(-\dfrac{34}{5}\right)\sqrt{63}\right)-\left(9-\left(\left(3\right)\sqrt{175}\right)\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{175}+\left(\dfrac{79}{8}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}-\dfrac{47}{9}\right)\right)\\
&=&\left(\dfrac{15}{7}\right)-\left(\left(\dfrac{455}{8}-\dfrac{231}{4}+\left(-\dfrac{102}{5}\right)\sqrt{7}\right)-\left(9-\left(\left(15\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{205}{2}\right)\sqrt{7}+\left(\dfrac{395}{8}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{180}{7}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}-\dfrac{47}{9}\right)\right)\\
&=&\left(\dfrac{15}{7}\right)-\left(-\dfrac{335}{72}+\left(-\dfrac{9891}{40}\right)\sqrt{7}\right)\\
&=&\dfrac{15}{7}+\dfrac{335}{72}+\left(\dfrac{9891}{40}\right)\sqrt{7}\\
&=&\dfrac{3425}{504}+\left(\dfrac{9891}{40}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{15}{7}\right)\times\left(\left(\left(\dfrac{65}{8}\right)\sqrt{49}+\left(-\dfrac{33}{4}\right)\sqrt{49}+\left(-\dfrac{34}{5}\right)\sqrt{63}\right)-\left(9-\left(\left(3\right)\sqrt{175}\right)\right)-\left(\left(\dfrac{41}{2}\right)\sqrt{175}+\left(\dfrac{79}{8}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{7}\right)\sqrt{175}+\left(\dfrac{81}{7}\right)\sqrt{175}-\dfrac{47}{9}\right)\right)\\
&=&\left(\dfrac{15}{7}\right)\times\left(\left(\dfrac{455}{8}-\dfrac{231}{4}+\left(-\dfrac{102}{5}\right)\sqrt{7}\right)-\left(9-\left(\left(15\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{205}{2}\right)\sqrt{7}+\left(\dfrac{395}{8}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{180}{7}\right)\sqrt{7}+\left(\dfrac{405}{7}\right)\sqrt{7}-\dfrac{47}{9}\right)\right)\\
&=&\left(\dfrac{15}{7}\right)\left(-\dfrac{335}{72}+\left(-\dfrac{9891}{40}\right)\sqrt{7}\right)\\
&=&-\dfrac{1675}{168}+\left(-\dfrac{4239}{8}\right)\sqrt{7}\\
&=&-\dfrac{1675}{168}+\left(-\dfrac{4239}{8}\right)\sqrt{7}\\
\end{eqnarray*}