L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{11}{4}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{49}\right)-\dfrac{5}{8}\) et \( Y=\left(\left(-\dfrac{67}{4}\right)\sqrt{63}+\left(\dfrac{28}{5}\right)\sqrt{63}\right)-\left(\left(5\right)\sqrt{49}-\dfrac{63}{5}+\dfrac{27}{8}+\left(-8\right)\sqrt{49}\right)\) . 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{11}{4}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{49}\right)-\dfrac{5}{8}\right)+\left(\left(\left(-\dfrac{67}{4}\right)\sqrt{63}+\left(\dfrac{28}{5}\right)\sqrt{63}\right)-\left(\left(5\right)\sqrt{49}-\dfrac{63}{5}+\dfrac{27}{8}+\left(-8\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(\dfrac{55}{4}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-79+\dfrac{119}{2}-\dfrac{5}{8}\right)+\left(\left(\left(-\dfrac{201}{4}\right)\sqrt{7}+\left(\dfrac{84}{5}\right)\sqrt{7}\right)-\left(35-\dfrac{63}{5}+\dfrac{27}{8}-56\right)\right)\\
&=&\left(\left(\dfrac{55}{4}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-79+\dfrac{119}{2}-\dfrac{5}{8}+\left(\left(-\dfrac{201}{4}\right)\sqrt{7}+\left(\dfrac{84}{5}\right)\sqrt{7}\right)-\left(35-\dfrac{63}{5}+\dfrac{27}{8}-56\right)\\
&=&\left(\dfrac{43}{10}\right)\sqrt{7}+\dfrac{101}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{11}{4}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{49}\right)-\dfrac{5}{8}\right)-\left(\left(\left(-\dfrac{67}{4}\right)\sqrt{63}+\left(\dfrac{28}{5}\right)\sqrt{63}\right)-\left(\left(5\right)\sqrt{49}-\dfrac{63}{5}+\dfrac{27}{8}+\left(-8\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(\dfrac{55}{4}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-79+\dfrac{119}{2}-\dfrac{5}{8}\right)-\left(\left(\left(-\dfrac{201}{4}\right)\sqrt{7}+\left(\dfrac{84}{5}\right)\sqrt{7}\right)-\left(35-\dfrac{63}{5}+\dfrac{27}{8}-56\right)\right)\\
&=&\left(\left(\dfrac{151}{4}\right)\sqrt{7}-\dfrac{161}{8}\right)-\left(\left(-\dfrac{669}{20}\right)\sqrt{7}+\dfrac{1209}{40}\right)\\
&=&\left(\dfrac{151}{4}\right)\sqrt{7}-\dfrac{161}{8}+\left(\dfrac{669}{20}\right)\sqrt{7}-\dfrac{1209}{40}\\
&=&\left(\dfrac{356}{5}\right)\sqrt{7}-\dfrac{1007}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{11}{4}\right)\sqrt{175}\right)-\left(\left(-8\right)\sqrt{63}\right)-\left(\left(\dfrac{79}{7}\right)\sqrt{49}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{49}\right)-\dfrac{5}{8}\right)\times\left(\left(\left(-\dfrac{67}{4}\right)\sqrt{63}+\left(\dfrac{28}{5}\right)\sqrt{63}\right)-\left(\left(5\right)\sqrt{49}-\dfrac{63}{5}+\dfrac{27}{8}+\left(-8\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(\dfrac{55}{4}\right)\sqrt{7}\right)-\left(\left(-24\right)\sqrt{7}\right)-79+\dfrac{119}{2}-\dfrac{5}{8}\right)\times\left(\left(\left(-\dfrac{201}{4}\right)\sqrt{7}+\left(\dfrac{84}{5}\right)\sqrt{7}\right)-\left(35-\dfrac{63}{5}+\dfrac{27}{8}-56\right)\right)\\
&=&\left(\left(\dfrac{151}{4}\right)\sqrt{7}-\dfrac{161}{8}\right)\left(\left(-\dfrac{669}{20}\right)\sqrt{7}+\dfrac{1209}{40}\right)\\
&=&\left(-\dfrac{101019}{80}\right)\sqrt{49}+\left(\dfrac{72567}{40}\right)\sqrt{7}-\dfrac{194649}{320}\\
&=&-\dfrac{3023181}{320}+\left(\dfrac{72567}{40}\right)\sqrt{7}\\
\end{eqnarray*}