L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)-\left(\left(5\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{27}\right)\right)\) et \( Y=\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(\dfrac{45}{8}\right)\sqrt{9}+\left(-\dfrac{21}{2}\right)\sqrt{12}+\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-5\) . 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{75}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)-\left(\left(5\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{27}\right)\right)\right)+\left(\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(\dfrac{45}{8}\right)\sqrt{9}+\left(-\dfrac{21}{2}\right)\sqrt{12}+\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-5\right)\\
&=&\left(\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{111}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)-\left(\left(15\right)\sqrt{3}\right)-\left(44-\left(\left(-\dfrac{145}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{123}{4}\right)\sqrt{3}\right)\right)\right)+\left(-38+\dfrac{135}{8}+\left(-21\right)\sqrt{3}+\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-5\right)\\
&=&\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{111}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)-\left(\left(15\right)\sqrt{3}\right)-\left(44-\left(\left(-\dfrac{145}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{123}{4}\right)\sqrt{3}\right)\right)-38+\dfrac{135}{8}+\left(-21\right)\sqrt{3}+\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-5\\
&=&\left(-\dfrac{110567}{504}\right)\sqrt{3}-\dfrac{561}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)-\left(\left(5\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{27}\right)\right)\right)-\left(\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(\dfrac{45}{8}\right)\sqrt{9}+\left(-\dfrac{21}{2}\right)\sqrt{12}+\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-5\right)\\
&=&\left(\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{111}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)-\left(\left(15\right)\sqrt{3}\right)-\left(44-\left(\left(-\dfrac{145}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{123}{4}\right)\sqrt{3}\right)\right)\right)-\left(-38+\dfrac{135}{8}+\left(-21\right)\sqrt{3}+\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-5\right)\\
&=&\left(\left(-\dfrac{117875}{504}\right)\sqrt{3}-44\right)-\left(-\dfrac{209}{8}+\left(\dfrac{29}{2}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{117875}{504}\right)\sqrt{3}-44+\dfrac{209}{8}+\left(-\dfrac{29}{2}\right)\sqrt{3}\\
&=&\left(-\dfrac{125183}{504}\right)\sqrt{3}-\dfrac{143}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{37}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{15}{8}\right)\sqrt{75}\right)-\left(\left(0\right)\sqrt{27}\right)\right)-\left(\left(5\right)\sqrt{27}\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{29}{9}\right)\sqrt{75}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{27}\right)\right)\right)\times\left(\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(\dfrac{45}{8}\right)\sqrt{9}+\left(-\dfrac{21}{2}\right)\sqrt{12}+\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-5\right)\\
&=&\left(\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{111}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{375}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)\right)-\left(\left(15\right)\sqrt{3}\right)-\left(44-\left(\left(-\dfrac{145}{9}\right)\sqrt{3}\right)-\left(\left(-\dfrac{123}{4}\right)\sqrt{3}\right)\right)\right)\times\left(-38+\dfrac{135}{8}+\left(-21\right)\sqrt{3}+\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-5\right)\\
&=&\left(\left(-\dfrac{117875}{504}\right)\sqrt{3}-44\right)\left(-\dfrac{209}{8}+\left(\dfrac{29}{2}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{22063459}{4032}\right)\sqrt{3}+\left(-\dfrac{3418375}{1008}\right)\sqrt{9}+\dfrac{2299}{2}\\
&=&\left(\dfrac{22063459}{4032}\right)\sqrt{3}-\dfrac{3032143}{336}\\
\end{eqnarray*}