L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\dfrac{45}{2}+\left(\dfrac{1}{3}\right)\sqrt{175}-5-\dfrac{78}{5}\) et \( Y=1-\left(\left(-\dfrac{34}{7}\right)\sqrt{175}+\left(-\dfrac{62}{3}\right)\sqrt{28}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}+\left(-\dfrac{17}{3}\right)\sqrt{63}+\left(8\right)\sqrt{28}\right)\) . 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(-\dfrac{56}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\dfrac{45}{2}+\left(\dfrac{1}{3}\right)\sqrt{175}-5-\dfrac{78}{5}\right)+\left(1-\left(\left(-\dfrac{34}{7}\right)\sqrt{175}+\left(-\dfrac{62}{3}\right)\sqrt{28}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}+\left(-\dfrac{17}{3}\right)\sqrt{63}+\left(8\right)\sqrt{28}\right)\right)\\
&=&\left(\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\dfrac{45}{2}+\left(\dfrac{5}{3}\right)\sqrt{7}-5-\dfrac{78}{5}\right)+\left(1-\left(\left(-\dfrac{170}{7}\right)\sqrt{7}+\left(-\dfrac{124}{3}\right)\sqrt{7}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}+\left(-17\right)\sqrt{7}+\left(16\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\dfrac{45}{2}+\left(\dfrac{5}{3}\right)\sqrt{7}-5-\dfrac{78}{5}+1-\left(\left(-\dfrac{170}{7}\right)\sqrt{7}+\left(-\dfrac{124}{3}\right)\sqrt{7}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}+\left(-17\right)\sqrt{7}+\left(16\right)\sqrt{7}\right)\\
&=&\left(\dfrac{1797}{14}\right)\sqrt{7}-\dfrac{777}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\dfrac{45}{2}+\left(\dfrac{1}{3}\right)\sqrt{175}-5-\dfrac{78}{5}\right)-\left(1-\left(\left(-\dfrac{34}{7}\right)\sqrt{175}+\left(-\dfrac{62}{3}\right)\sqrt{28}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}+\left(-\dfrac{17}{3}\right)\sqrt{63}+\left(8\right)\sqrt{28}\right)\right)\\
&=&\left(\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\dfrac{45}{2}+\left(\dfrac{5}{3}\right)\sqrt{7}-5-\dfrac{78}{5}\right)-\left(1-\left(\left(-\dfrac{170}{7}\right)\sqrt{7}+\left(-\dfrac{124}{3}\right)\sqrt{7}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}+\left(-17\right)\sqrt{7}+\left(16\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{950}{21}\right)\sqrt{7}-\dfrac{431}{10}\right)-\left(\dfrac{17}{4}+\left(\dfrac{3491}{42}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{950}{21}\right)\sqrt{7}-\dfrac{431}{10}+-\dfrac{17}{4}+\left(-\dfrac{3491}{42}\right)\sqrt{7}\\
&=&\left(-\dfrac{1591}{42}\right)\sqrt{7}-\dfrac{947}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{175}\right)-\left(\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\dfrac{45}{2}+\left(\dfrac{1}{3}\right)\sqrt{175}-5-\dfrac{78}{5}\right)\times\left(1-\left(\left(-\dfrac{34}{7}\right)\sqrt{175}+\left(-\dfrac{62}{3}\right)\sqrt{28}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}+\left(-\dfrac{17}{3}\right)\sqrt{63}+\left(8\right)\sqrt{28}\right)\right)\\
&=&\left(\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\dfrac{45}{2}+\left(\dfrac{5}{3}\right)\sqrt{7}-5-\dfrac{78}{5}\right)\times\left(1-\left(\left(-\dfrac{170}{7}\right)\sqrt{7}+\left(-\dfrac{124}{3}\right)\sqrt{7}-\dfrac{13}{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}+\left(-17\right)\sqrt{7}+\left(16\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{950}{21}\right)\sqrt{7}-\dfrac{431}{10}\right)\left(\dfrac{17}{4}+\left(\dfrac{3491}{42}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{1423871}{420}\right)\sqrt{7}+\left(\dfrac{1658225}{441}\right)\sqrt{49}-\dfrac{7327}{40}\\
&=&\left(-\dfrac{1423871}{420}\right)\sqrt{7}+\dfrac{65867399}{2520}\\
\end{eqnarray*}