L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{44}{7}\right)\sqrt{27}+\left(-\dfrac{74}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{12}\right)+8\) et \( Y=\left(\left(3\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{9}+\left(\dfrac{32}{9}\right)\sqrt{75}+\left(-\dfrac{5}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{7}{8}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\) . 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{44}{7}\right)\sqrt{27}+\left(-\dfrac{74}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{12}\right)+8\right)+\left(\left(\left(3\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{9}+\left(\dfrac{32}{9}\right)\sqrt{75}+\left(-\dfrac{5}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{7}{8}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(-\dfrac{132}{7}\right)\sqrt{3}+\left(-74\right)\sqrt{3}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{3}\right)+8\right)+\left(9-\left(\dfrac{63}{2}+\left(\dfrac{160}{9}\right)\sqrt{3}-\dfrac{15}{2}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{3}\right)-0\right)\\
&=&\left(\left(-\dfrac{132}{7}\right)\sqrt{3}+\left(-74\right)\sqrt{3}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{3}\right)+8+9-\left(\dfrac{63}{2}+\left(\dfrac{160}{9}\right)\sqrt{3}-\dfrac{15}{2}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{3}\right)-0\\
&=&\left(-\dfrac{25423}{252}\right)\sqrt{3}-7\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{27}+\left(-\dfrac{74}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{12}\right)+8\right)-\left(\left(\left(3\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{9}+\left(\dfrac{32}{9}\right)\sqrt{75}+\left(-\dfrac{5}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{7}{8}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(-\dfrac{132}{7}\right)\sqrt{3}+\left(-74\right)\sqrt{3}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{3}\right)+8\right)-\left(9-\left(\dfrac{63}{2}+\left(\dfrac{160}{9}\right)\sqrt{3}-\dfrac{15}{2}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{3}\right)-0\right)\\
&=&\left(\left(-\dfrac{1139}{14}\right)\sqrt{3}+8\right)-\left(-15+\left(-\dfrac{703}{36}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{1139}{14}\right)\sqrt{3}+8+15+\left(\dfrac{703}{36}\right)\sqrt{3}\\
&=&\left(-\dfrac{15581}{252}\right)\sqrt{3}+23\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{27}+\left(-\dfrac{74}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{4}\right)\sqrt{12}\right)+8\right)\times\left(\left(\left(3\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{9}+\left(\dfrac{32}{9}\right)\sqrt{75}+\left(-\dfrac{5}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{7}{8}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(-\dfrac{132}{7}\right)\sqrt{3}+\left(-74\right)\sqrt{3}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{3}\right)+8\right)\times\left(9-\left(\dfrac{63}{2}+\left(\dfrac{160}{9}\right)\sqrt{3}-\dfrac{15}{2}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{3}\right)-0\right)\\
&=&\left(\left(-\dfrac{1139}{14}\right)\sqrt{3}+8\right)\left(-15+\left(-\dfrac{703}{36}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{134081}{126}\right)\sqrt{3}+\left(\dfrac{800717}{504}\right)\sqrt{9}-120\\
&=&\left(\dfrac{134081}{126}\right)\sqrt{3}+\dfrac{780557}{168}\\
\end{eqnarray*}