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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{17}{4}\right)\sqrt{9}+\left(\dfrac{56}{3}\right)\sqrt{9}+\left(-\dfrac{58}{7}\right)\sqrt{75}+\dfrac{48}{7}-\dfrac{59}{9}\) et \( Y=-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{4}\right)\sqrt{12}+0+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{9}\right)\sqrt{9}+\left(-\dfrac{77}{8}\right)\sqrt{27}+\left(\dfrac{27}{4}\right)\sqrt{12}+\dfrac{37}{2}+\left(-\dfrac{28}{5}\right)\sqrt{27}+\left(\dfrac{35}{2}\right)\sqrt{9}\) . 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{17}{4}\right)\sqrt{9}+\left(\dfrac{56}{3}\right)\sqrt{9}+\left(-\dfrac{58}{7}\right)\sqrt{75}+\dfrac{48}{7}-\dfrac{59}{9}\right)+\left(-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{4}\right)\sqrt{12}+0+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{9}\right)\sqrt{9}+\left(-\dfrac{77}{8}\right)\sqrt{27}+\left(\dfrac{27}{4}\right)\sqrt{12}+\dfrac{37}{2}+\left(-\dfrac{28}{5}\right)\sqrt{27}+\left(\dfrac{35}{2}\right)\sqrt{9}\right)\\
&=&\left(\dfrac{51}{4}+56+\left(-\dfrac{290}{7}\right)\sqrt{3}+\dfrac{48}{7}-\dfrac{59}{9}\right)+\left(-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{2}\right)\sqrt{3}+0+\left(0\right)\sqrt{3}-\dfrac{47}{3}+\left(-\dfrac{231}{8}\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}+\dfrac{37}{2}+\left(-\dfrac{84}{5}\right)\sqrt{3}+\dfrac{105}{2}\right)\\
&=&\dfrac{51}{4}+56+\left(-\dfrac{290}{7}\right)\sqrt{3}+\dfrac{48}{7}-\dfrac{59}{9}-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{2}\right)\sqrt{3}+0+\left(0\right)\sqrt{3}-\dfrac{47}{3}+\left(-\dfrac{231}{8}\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}+\dfrac{37}{2}+\left(-\dfrac{84}{5}\right)\sqrt{3}+\dfrac{105}{2}\\
&=&\dfrac{15389}{126}+\left(-\dfrac{24389}{280}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{17}{4}\right)\sqrt{9}+\left(\dfrac{56}{3}\right)\sqrt{9}+\left(-\dfrac{58}{7}\right)\sqrt{75}+\dfrac{48}{7}-\dfrac{59}{9}\right)-\left(-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{4}\right)\sqrt{12}+0+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{9}\right)\sqrt{9}+\left(-\dfrac{77}{8}\right)\sqrt{27}+\left(\dfrac{27}{4}\right)\sqrt{12}+\dfrac{37}{2}+\left(-\dfrac{28}{5}\right)\sqrt{27}+\left(\dfrac{35}{2}\right)\sqrt{9}\right)\\
&=&\left(\dfrac{51}{4}+56+\left(-\dfrac{290}{7}\right)\sqrt{3}+\dfrac{48}{7}-\dfrac{59}{9}\right)-\left(-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{2}\right)\sqrt{3}+0+\left(0\right)\sqrt{3}-\dfrac{47}{3}+\left(-\dfrac{231}{8}\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}+\dfrac{37}{2}+\left(-\dfrac{84}{5}\right)\sqrt{3}+\dfrac{105}{2}\right)\\
&=&\left(\dfrac{17401}{252}+\left(-\dfrac{290}{7}\right)\sqrt{3}\right)-\left(\dfrac{637}{12}+\left(-\dfrac{1827}{40}\right)\sqrt{3}\right)\\
&=&\dfrac{17401}{252}+\left(-\dfrac{290}{7}\right)\sqrt{3}+-\dfrac{637}{12}+\left(\dfrac{1827}{40}\right)\sqrt{3}\\
&=&\dfrac{1006}{63}+\left(\dfrac{1189}{280}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{17}{4}\right)\sqrt{9}+\left(\dfrac{56}{3}\right)\sqrt{9}+\left(-\dfrac{58}{7}\right)\sqrt{75}+\dfrac{48}{7}-\dfrac{59}{9}\right)\times\left(-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{4}\right)\sqrt{12}+0+\left(0\right)\sqrt{27}+\left(-\dfrac{47}{9}\right)\sqrt{9}+\left(-\dfrac{77}{8}\right)\sqrt{27}+\left(\dfrac{27}{4}\right)\sqrt{12}+\dfrac{37}{2}+\left(-\dfrac{28}{5}\right)\sqrt{27}+\left(\dfrac{35}{2}\right)\sqrt{9}\right)\\
&=&\left(\dfrac{51}{4}+56+\left(-\dfrac{290}{7}\right)\sqrt{3}+\dfrac{48}{7}-\dfrac{59}{9}\right)\times\left(-\dfrac{15}{2}+\dfrac{21}{4}+\left(-\dfrac{27}{2}\right)\sqrt{3}+0+\left(0\right)\sqrt{3}-\dfrac{47}{3}+\left(-\dfrac{231}{8}\right)\sqrt{3}+\left(\dfrac{27}{2}\right)\sqrt{3}+\dfrac{37}{2}+\left(-\dfrac{84}{5}\right)\sqrt{3}+\dfrac{105}{2}\right)\\
&=&\left(\dfrac{17401}{252}+\left(-\dfrac{290}{7}\right)\sqrt{3}\right)\left(\dfrac{637}{12}+\left(-\dfrac{1827}{40}\right)\sqrt{3}\right)\\
&=&\dfrac{1583491}{432}+\left(-\dfrac{2569487}{480}\right)\sqrt{3}+\left(\dfrac{7569}{4}\right)\sqrt{9}\\
&=&\dfrac{4035847}{432}+\left(-\dfrac{2569487}{480}\right)\sqrt{3}\\
\end{eqnarray*}