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Exercice
Soit \( X=\left(\dfrac{3}{2}\right)\sqrt{75}+\left(-\dfrac{23}{6}\right)\sqrt{12}+\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\dfrac{77}{3}\right)\sqrt{9}+\left(-\dfrac{25}{2}\right)\sqrt{27}+\left(\dfrac{29}{9}\right)\sqrt{12}+\left(\dfrac{71}{2}\right)\sqrt{75}+\dfrac{29}{2}+\left(\dfrac{81}{2}\right)\sqrt{27}\) et \( Y=\left(-\dfrac{4}{3}\right)\sqrt{12}+\left(-\dfrac{45}{4}\right)\sqrt{75}+\left(\dfrac{29}{2}\right)\sqrt{75}+\left(\dfrac{9}{2}\right)\sqrt{75}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{4}\right)\sqrt{12}\right)+\left(\dfrac{7}{3}\right)\sqrt{9}\) . 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{3}{2}\right)\sqrt{75}+\left(-\dfrac{23}{6}\right)\sqrt{12}+\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\dfrac{77}{3}\right)\sqrt{9}+\left(-\dfrac{25}{2}\right)\sqrt{27}+\left(\dfrac{29}{9}\right)\sqrt{12}+\left(\dfrac{71}{2}\right)\sqrt{75}+\dfrac{29}{2}+\left(\dfrac{81}{2}\right)\sqrt{27}\right)+\left(\left(-\dfrac{4}{3}\right)\sqrt{12}+\left(-\dfrac{45}{4}\right)\sqrt{75}+\left(\dfrac{29}{2}\right)\sqrt{75}+\left(\dfrac{9}{2}\right)\sqrt{75}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{4}\right)\sqrt{12}\right)+\left(\dfrac{7}{3}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+\left(-\dfrac{23}{3}\right)\sqrt{3}+\left(\dfrac{15}{2}\right)\sqrt{3}+77+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(\dfrac{58}{9}\right)\sqrt{3}+\left(\dfrac{355}{2}\right)\sqrt{3}+\dfrac{29}{2}+\left(\dfrac{243}{2}\right)\sqrt{3}\right)+\left(\left(-\dfrac{8}{3}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}+\left(\dfrac{145}{2}\right)\sqrt{3}+\left(\dfrac{45}{2}\right)\sqrt{3}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{2}\right)\sqrt{3}\right)+7\right)\\
&=&\left(\dfrac{15}{2}\right)\sqrt{3}+\left(-\dfrac{23}{3}\right)\sqrt{3}+\left(\dfrac{15}{2}\right)\sqrt{3}+77+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(\dfrac{58}{9}\right)\sqrt{3}+\left(\dfrac{355}{2}\right)\sqrt{3}+\dfrac{29}{2}+\left(\dfrac{243}{2}\right)\sqrt{3}+\left(-\dfrac{8}{3}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}+\left(\dfrac{145}{2}\right)\sqrt{3}+\left(\dfrac{45}{2}\right)\sqrt{3}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{2}\right)\sqrt{3}\right)+7\\
&=&\left(\dfrac{12163}{36}\right)\sqrt{3}+\dfrac{20369}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{3}{2}\right)\sqrt{75}+\left(-\dfrac{23}{6}\right)\sqrt{12}+\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\dfrac{77}{3}\right)\sqrt{9}+\left(-\dfrac{25}{2}\right)\sqrt{27}+\left(\dfrac{29}{9}\right)\sqrt{12}+\left(\dfrac{71}{2}\right)\sqrt{75}+\dfrac{29}{2}+\left(\dfrac{81}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{12}+\left(-\dfrac{45}{4}\right)\sqrt{75}+\left(\dfrac{29}{2}\right)\sqrt{75}+\left(\dfrac{9}{2}\right)\sqrt{75}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{4}\right)\sqrt{12}\right)+\left(\dfrac{7}{3}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+\left(-\dfrac{23}{3}\right)\sqrt{3}+\left(\dfrac{15}{2}\right)\sqrt{3}+77+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(\dfrac{58}{9}\right)\sqrt{3}+\left(\dfrac{355}{2}\right)\sqrt{3}+\dfrac{29}{2}+\left(\dfrac{243}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}+\left(\dfrac{145}{2}\right)\sqrt{3}+\left(\dfrac{45}{2}\right)\sqrt{3}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{2}\right)\sqrt{3}\right)+7\right)\\
&=&\left(\left(\dfrac{4955}{18}\right)\sqrt{3}+\dfrac{183}{2}\right)-\left(\left(\dfrac{751}{12}\right)\sqrt{3}+\dfrac{3899}{180}\right)\\
&=&\left(\dfrac{4955}{18}\right)\sqrt{3}+\dfrac{183}{2}+\left(-\dfrac{751}{12}\right)\sqrt{3}-\dfrac{3899}{180}\\
&=&\left(\dfrac{7657}{36}\right)\sqrt{3}+\dfrac{12571}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{3}{2}\right)\sqrt{75}+\left(-\dfrac{23}{6}\right)\sqrt{12}+\left(\dfrac{3}{2}\right)\sqrt{75}+\left(\dfrac{77}{3}\right)\sqrt{9}+\left(-\dfrac{25}{2}\right)\sqrt{27}+\left(\dfrac{29}{9}\right)\sqrt{12}+\left(\dfrac{71}{2}\right)\sqrt{75}+\dfrac{29}{2}+\left(\dfrac{81}{2}\right)\sqrt{27}\right)\times\left(\left(-\dfrac{4}{3}\right)\sqrt{12}+\left(-\dfrac{45}{4}\right)\sqrt{75}+\left(\dfrac{29}{2}\right)\sqrt{75}+\left(\dfrac{9}{2}\right)\sqrt{75}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{4}\right)\sqrt{12}\right)+\left(\dfrac{7}{3}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{15}{2}\right)\sqrt{3}+\left(-\dfrac{23}{3}\right)\sqrt{3}+\left(\dfrac{15}{2}\right)\sqrt{3}+77+\left(-\dfrac{75}{2}\right)\sqrt{3}+\left(\dfrac{58}{9}\right)\sqrt{3}+\left(\dfrac{355}{2}\right)\sqrt{3}+\dfrac{29}{2}+\left(\dfrac{243}{2}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{8}{3}\right)\sqrt{3}+\left(-\dfrac{225}{4}\right)\sqrt{3}+\left(\dfrac{145}{2}\right)\sqrt{3}+\left(\dfrac{45}{2}\right)\sqrt{3}-\dfrac{8}{9}+\dfrac{74}{5}+\dfrac{3}{4}-\left(\left(-\dfrac{53}{2}\right)\sqrt{3}\right)+7\right)\\
&=&\left(\left(\dfrac{4955}{18}\right)\sqrt{3}+\dfrac{183}{2}\right)\left(\left(\dfrac{751}{12}\right)\sqrt{3}+\dfrac{3899}{180}\right)\\
&=&\left(\dfrac{3721205}{216}\right)\sqrt{9}+\left(\dfrac{946825}{81}\right)\sqrt{3}+\dfrac{237839}{120}\\
&=&\dfrac{9659771}{180}+\left(\dfrac{946825}{81}\right)\sqrt{3}\\
\end{eqnarray*}