L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{48}{7}\right)\sqrt{28}+\left(-\dfrac{73}{6}\right)\sqrt{49}+\left(\dfrac{71}{7}\right)\sqrt{63}-\dfrac{63}{5}+\left(2\right)\sqrt{175}\right)-\left(\dfrac{69}{5}+\left(-4\right)\sqrt{28}+\left(\dfrac{77}{2}\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\) et \( Y=\left(3\right)\sqrt{49}+\dfrac{27}{2}\) . 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{48}{7}\right)\sqrt{28}+\left(-\dfrac{73}{6}\right)\sqrt{49}+\left(\dfrac{71}{7}\right)\sqrt{63}-\dfrac{63}{5}+\left(2\right)\sqrt{175}\right)-\left(\dfrac{69}{5}+\left(-4\right)\sqrt{28}+\left(\dfrac{77}{2}\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\right)+\left(\left(3\right)\sqrt{49}+\dfrac{27}{2}\right)\\
&=&\left(\left(\left(-\dfrac{96}{7}\right)\sqrt{7}-\dfrac{511}{6}+\left(\dfrac{213}{7}\right)\sqrt{7}-\dfrac{63}{5}+\left(10\right)\sqrt{7}\right)-\left(\dfrac{69}{5}+\left(-8\right)\sqrt{7}+\dfrac{539}{2}+\left(-18\right)\sqrt{7}\right)\right)+\left(21+\dfrac{27}{2}\right)\\
&=&\left(\left(-\dfrac{96}{7}\right)\sqrt{7}-\dfrac{511}{6}+\left(\dfrac{213}{7}\right)\sqrt{7}-\dfrac{63}{5}+\left(10\right)\sqrt{7}\right)-\left(\dfrac{69}{5}+\left(-8\right)\sqrt{7}+\dfrac{539}{2}+\left(-18\right)\sqrt{7}\right)+21+\dfrac{27}{2}\\
&=&\left(\dfrac{369}{7}\right)\sqrt{7}-\dfrac{10397}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{48}{7}\right)\sqrt{28}+\left(-\dfrac{73}{6}\right)\sqrt{49}+\left(\dfrac{71}{7}\right)\sqrt{63}-\dfrac{63}{5}+\left(2\right)\sqrt{175}\right)-\left(\dfrac{69}{5}+\left(-4\right)\sqrt{28}+\left(\dfrac{77}{2}\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\right)-\left(\left(3\right)\sqrt{49}+\dfrac{27}{2}\right)\\
&=&\left(\left(\left(-\dfrac{96}{7}\right)\sqrt{7}-\dfrac{511}{6}+\left(\dfrac{213}{7}\right)\sqrt{7}-\dfrac{63}{5}+\left(10\right)\sqrt{7}\right)-\left(\dfrac{69}{5}+\left(-8\right)\sqrt{7}+\dfrac{539}{2}+\left(-18\right)\sqrt{7}\right)\right)-\left(21+\dfrac{27}{2}\right)\\
&=&\left(\left(\dfrac{369}{7}\right)\sqrt{7}-\dfrac{5716}{15}\right)-\left(\dfrac{69}{2}\right)\\
&=&\left(\dfrac{369}{7}\right)\sqrt{7}-\dfrac{5716}{15}+-\dfrac{69}{2}\\
&=&\left(\dfrac{369}{7}\right)\sqrt{7}-\dfrac{12467}{30}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{48}{7}\right)\sqrt{28}+\left(-\dfrac{73}{6}\right)\sqrt{49}+\left(\dfrac{71}{7}\right)\sqrt{63}-\dfrac{63}{5}+\left(2\right)\sqrt{175}\right)-\left(\dfrac{69}{5}+\left(-4\right)\sqrt{28}+\left(\dfrac{77}{2}\right)\sqrt{49}+\left(-9\right)\sqrt{28}\right)\right)\times\left(\left(3\right)\sqrt{49}+\dfrac{27}{2}\right)\\
&=&\left(\left(\left(-\dfrac{96}{7}\right)\sqrt{7}-\dfrac{511}{6}+\left(\dfrac{213}{7}\right)\sqrt{7}-\dfrac{63}{5}+\left(10\right)\sqrt{7}\right)-\left(\dfrac{69}{5}+\left(-8\right)\sqrt{7}+\dfrac{539}{2}+\left(-18\right)\sqrt{7}\right)\right)\times\left(21+\dfrac{27}{2}\right)\\
&=&\left(\left(\dfrac{369}{7}\right)\sqrt{7}-\dfrac{5716}{15}\right)\left(\dfrac{69}{2}\right)\\
&=&\left(\dfrac{25461}{14}\right)\sqrt{7}-\dfrac{65734}{5}\\
&=&\left(\dfrac{25461}{14}\right)\sqrt{7}-\dfrac{65734}{5}\\
\end{eqnarray*}