L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{29}{6}\right)\sqrt{4}\) et \( Y=\left(\dfrac{28}{3}\right)\sqrt{4}+\left(-\dfrac{4}{7}\right)\sqrt{8}+\left(\dfrac{29}{7}\right)\sqrt{8}+\left(\dfrac{49}{3}\right)\sqrt{50}+\left(\left(\dfrac{29}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{76}{9}\right)\sqrt{4}\right)-\dfrac{7}{5}-\left(\left(-\dfrac{52}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{8}\right)\) . 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{29}{6}\right)\sqrt{4}\right)+\left(\left(\dfrac{28}{3}\right)\sqrt{4}+\left(-\dfrac{4}{7}\right)\sqrt{8}+\left(\dfrac{29}{7}\right)\sqrt{8}+\left(\dfrac{49}{3}\right)\sqrt{50}+\left(\left(\dfrac{29}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{76}{9}\right)\sqrt{4}\right)-\dfrac{7}{5}-\left(\left(-\dfrac{52}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{8}\right)\right)\\
&=&\left(\dfrac{29}{3}\right)+\left(\dfrac{56}{3}+\left(-\dfrac{8}{7}\right)\sqrt{2}+\left(\dfrac{58}{7}\right)\sqrt{2}+\left(\dfrac{245}{3}\right)\sqrt{2}+\left(\left(\dfrac{58}{7}\right)\sqrt{2}\right)-\dfrac{152}{9}-\dfrac{7}{5}+\dfrac{104}{9}-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\right)\\
&=&\dfrac{29}{3}+\dfrac{56}{3}+\left(-\dfrac{8}{7}\right)\sqrt{2}+\left(\dfrac{58}{7}\right)\sqrt{2}+\left(\dfrac{245}{3}\right)\sqrt{2}+\left(\left(\dfrac{58}{7}\right)\sqrt{2}\right)-\dfrac{152}{9}-\dfrac{7}{5}+\dfrac{104}{9}-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\\
&=&\dfrac{108}{5}+\left(\dfrac{2369}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{29}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{4}+\left(-\dfrac{4}{7}\right)\sqrt{8}+\left(\dfrac{29}{7}\right)\sqrt{8}+\left(\dfrac{49}{3}\right)\sqrt{50}+\left(\left(\dfrac{29}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{76}{9}\right)\sqrt{4}\right)-\dfrac{7}{5}-\left(\left(-\dfrac{52}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{8}\right)\right)\\
&=&\left(\dfrac{29}{3}\right)-\left(\dfrac{56}{3}+\left(-\dfrac{8}{7}\right)\sqrt{2}+\left(\dfrac{58}{7}\right)\sqrt{2}+\left(\dfrac{245}{3}\right)\sqrt{2}+\left(\left(\dfrac{58}{7}\right)\sqrt{2}\right)-\dfrac{152}{9}-\dfrac{7}{5}+\dfrac{104}{9}-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{29}{3}\right)-\left(\dfrac{179}{15}+\left(\dfrac{2369}{21}\right)\sqrt{2}\right)\\
&=&\dfrac{29}{3}+-\dfrac{179}{15}+\left(-\dfrac{2369}{21}\right)\sqrt{2}\\
&=&-\dfrac{34}{15}+\left(-\dfrac{2369}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{29}{6}\right)\sqrt{4}\right)\times\left(\left(\dfrac{28}{3}\right)\sqrt{4}+\left(-\dfrac{4}{7}\right)\sqrt{8}+\left(\dfrac{29}{7}\right)\sqrt{8}+\left(\dfrac{49}{3}\right)\sqrt{50}+\left(\left(\dfrac{29}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{76}{9}\right)\sqrt{4}\right)-\dfrac{7}{5}-\left(\left(-\dfrac{52}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{8}\right)\right)\\
&=&\left(\dfrac{29}{3}\right)\times\left(\dfrac{56}{3}+\left(-\dfrac{8}{7}\right)\sqrt{2}+\left(\dfrac{58}{7}\right)\sqrt{2}+\left(\dfrac{245}{3}\right)\sqrt{2}+\left(\left(\dfrac{58}{7}\right)\sqrt{2}\right)-\dfrac{152}{9}-\dfrac{7}{5}+\dfrac{104}{9}-\left(\left(-\dfrac{110}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{29}{3}\right)\left(\dfrac{179}{15}+\left(\dfrac{2369}{21}\right)\sqrt{2}\right)\\
&=&\dfrac{5191}{45}+\left(\dfrac{68701}{63}\right)\sqrt{2}\\
&=&\dfrac{5191}{45}+\left(\dfrac{68701}{63}\right)\sqrt{2}\\
\end{eqnarray*}