L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{7}{3}\right)\sqrt{49}+7+\left(8\right)\sqrt{175}-2\) et \( Y=\left(\left(-\dfrac{71}{8}\right)\sqrt{63}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{17}{7}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}+\left(-\dfrac{58}{7}\right)\sqrt{175}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{61}{9}\right)\sqrt{175}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{7}{3}\right)\sqrt{49}+7+\left(8\right)\sqrt{175}-2\right)+\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{63}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{17}{7}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}+\left(-\dfrac{58}{7}\right)\sqrt{175}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{61}{9}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\dfrac{49}{3}+7+\left(40\right)\sqrt{7}-2\right)+\left(\left(\left(-\dfrac{213}{8}\right)\sqrt{7}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{85}{7}\right)\sqrt{7}\right)+56\right)-\left(\left(-13\right)\sqrt{7}+\left(-\dfrac{290}{7}\right)\sqrt{7}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{305}{9}\right)\sqrt{7}\right)\right)\right)\\
&=&\dfrac{49}{3}+7+\left(40\right)\sqrt{7}-2+\left(\left(-\dfrac{213}{8}\right)\sqrt{7}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{85}{7}\right)\sqrt{7}\right)+56\right)-\left(\left(-13\right)\sqrt{7}+\left(-\dfrac{290}{7}\right)\sqrt{7}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{305}{9}\right)\sqrt{7}\right)\right)\\
&=&-\dfrac{1376}{105}+\left(\dfrac{23213}{504}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{7}{3}\right)\sqrt{49}+7+\left(8\right)\sqrt{175}-2\right)-\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{63}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{17}{7}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}+\left(-\dfrac{58}{7}\right)\sqrt{175}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{61}{9}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\dfrac{49}{3}+7+\left(40\right)\sqrt{7}-2\right)-\left(\left(\left(-\dfrac{213}{8}\right)\sqrt{7}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{85}{7}\right)\sqrt{7}\right)+56\right)-\left(\left(-13\right)\sqrt{7}+\left(-\dfrac{290}{7}\right)\sqrt{7}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{305}{9}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\dfrac{64}{3}+\left(40\right)\sqrt{7}\right)-\left(\left(\dfrac{3053}{504}\right)\sqrt{7}-\dfrac{3616}{105}\right)\\
&=&\dfrac{64}{3}+\left(40\right)\sqrt{7}+\left(-\dfrac{3053}{504}\right)\sqrt{7}+\dfrac{3616}{105}\\
&=&\dfrac{1952}{35}+\left(\dfrac{17107}{504}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{7}{3}\right)\sqrt{49}+7+\left(8\right)\sqrt{175}-2\right)\times\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{63}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{17}{7}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}+\left(-\dfrac{58}{7}\right)\sqrt{175}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{61}{9}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\dfrac{49}{3}+7+\left(40\right)\sqrt{7}-2\right)\times\left(\left(\left(-\dfrac{213}{8}\right)\sqrt{7}\right)-\dfrac{60}{7}-\left(\dfrac{79}{6}-\left(\left(\dfrac{85}{7}\right)\sqrt{7}\right)+56\right)-\left(\left(-13\right)\sqrt{7}+\left(-\dfrac{290}{7}\right)\sqrt{7}-\dfrac{75}{2}\right)-\left(-\dfrac{29}{5}-\left(\left(-\dfrac{305}{9}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\dfrac{64}{3}+\left(40\right)\sqrt{7}\right)\left(\left(\dfrac{3053}{504}\right)\sqrt{7}-\dfrac{3616}{105}\right)\\
&=&\left(-\dfrac{33704}{27}\right)\sqrt{7}-\dfrac{231424}{315}+\left(\dfrac{15265}{63}\right)\sqrt{49}\\
&=&\left(-\dfrac{33704}{27}\right)\sqrt{7}+\dfrac{302851}{315}\\
\end{eqnarray*}