L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{45}{4}\right)\sqrt{49}+\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{35}{8}\right)\sqrt{28}+\left(4\right)\sqrt{49}+\left(\dfrac{11}{2}\right)\sqrt{49}+\left(-\dfrac{35}{6}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{28}+\left(-\dfrac{56}{5}\right)\sqrt{63}\) et \( Y=\left(-\dfrac{5}{2}\right)\sqrt{49}\) . 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{45}{4}\right)\sqrt{49}+\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{35}{8}\right)\sqrt{28}+\left(4\right)\sqrt{49}+\left(\dfrac{11}{2}\right)\sqrt{49}+\left(-\dfrac{35}{6}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{28}+\left(-\dfrac{56}{5}\right)\sqrt{63}\right)+\left(\left(-\dfrac{5}{2}\right)\sqrt{49}\right)\\
&=&\left(-\dfrac{315}{4}+\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{35}{4}\right)\sqrt{7}+28+\dfrac{77}{2}-\dfrac{245}{6}+\left(\dfrac{8}{3}\right)\sqrt{7}+\left(-\dfrac{168}{5}\right)\sqrt{7}\right)+\left(-\dfrac{35}{2}\right)\\
&=&-\dfrac{315}{4}+\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{35}{4}\right)\sqrt{7}+28+\dfrac{77}{2}-\dfrac{245}{6}+\left(\dfrac{8}{3}\right)\sqrt{7}+\left(-\dfrac{168}{5}\right)\sqrt{7}-\dfrac{35}{2}\\
&=&-\dfrac{847}{12}+\left(\dfrac{919}{60}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{45}{4}\right)\sqrt{49}+\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{35}{8}\right)\sqrt{28}+\left(4\right)\sqrt{49}+\left(\dfrac{11}{2}\right)\sqrt{49}+\left(-\dfrac{35}{6}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{28}+\left(-\dfrac{56}{5}\right)\sqrt{63}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{49}\right)\\
&=&\left(-\dfrac{315}{4}+\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{35}{4}\right)\sqrt{7}+28+\dfrac{77}{2}-\dfrac{245}{6}+\left(\dfrac{8}{3}\right)\sqrt{7}+\left(-\dfrac{168}{5}\right)\sqrt{7}\right)-\left(-\dfrac{35}{2}\right)\\
&=&\left(-\dfrac{637}{12}+\left(\dfrac{919}{60}\right)\sqrt{7}\right)-\left(-\dfrac{35}{2}\right)\\
&=&-\dfrac{637}{12}+\left(\dfrac{919}{60}\right)\sqrt{7}+\dfrac{35}{2}\\
&=&-\dfrac{427}{12}+\left(\dfrac{919}{60}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{45}{4}\right)\sqrt{49}+\left(\dfrac{25}{2}\right)\sqrt{63}+\left(\dfrac{35}{8}\right)\sqrt{28}+\left(4\right)\sqrt{49}+\left(\dfrac{11}{2}\right)\sqrt{49}+\left(-\dfrac{35}{6}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{28}+\left(-\dfrac{56}{5}\right)\sqrt{63}\right)\times\left(\left(-\dfrac{5}{2}\right)\sqrt{49}\right)\\
&=&\left(-\dfrac{315}{4}+\left(\dfrac{75}{2}\right)\sqrt{7}+\left(\dfrac{35}{4}\right)\sqrt{7}+28+\dfrac{77}{2}-\dfrac{245}{6}+\left(\dfrac{8}{3}\right)\sqrt{7}+\left(-\dfrac{168}{5}\right)\sqrt{7}\right)\times\left(-\dfrac{35}{2}\right)\\
&=&\left(-\dfrac{637}{12}+\left(\dfrac{919}{60}\right)\sqrt{7}\right)\left(-\dfrac{35}{2}\right)\\
&=&\dfrac{22295}{24}+\left(-\dfrac{6433}{24}\right)\sqrt{7}\\
&=&\dfrac{22295}{24}+\left(-\dfrac{6433}{24}\right)\sqrt{7}\\
\end{eqnarray*}