L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(5\right)\sqrt{25}-\dfrac{44}{3}+\dfrac{27}{4}+\left(5\right)\sqrt{20}+\left(\dfrac{7}{4}\right)\sqrt{125}+\dfrac{61}{5}-\left(\left(\dfrac{7}{4}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{20}\right)+\left(-\dfrac{27}{2}\right)\sqrt{25}\) et \( Y=\left(\dfrac{65}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{125}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{16}{9}\right)\sqrt{20}+\left(\left(-\dfrac{62}{9}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{45}\right)+\left(\dfrac{37}{8}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(5\right)\sqrt{25}-\dfrac{44}{3}+\dfrac{27}{4}+\left(5\right)\sqrt{20}+\left(\dfrac{7}{4}\right)\sqrt{125}+\dfrac{61}{5}-\left(\left(\dfrac{7}{4}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{20}\right)+\left(-\dfrac{27}{2}\right)\sqrt{25}\right)+\left(\left(\dfrac{65}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{125}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{16}{9}\right)\sqrt{20}+\left(\left(-\dfrac{62}{9}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{45}\right)+\left(\dfrac{37}{8}\right)\sqrt{20}\right)\\
&=&\left(25-\dfrac{44}{3}+\dfrac{27}{4}+\left(10\right)\sqrt{5}+\left(\dfrac{35}{4}\right)\sqrt{5}+\dfrac{61}{5}-\left(\left(\dfrac{35}{4}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{5}\right)-\dfrac{135}{2}\right)+\left(\left(\dfrac{325}{9}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{32}{9}\right)\sqrt{5}+\left(\left(-\dfrac{124}{9}\right)\sqrt{5}\right)-\left(\left(-9\right)\sqrt{5}\right)+\left(\dfrac{37}{4}\right)\sqrt{5}\right)\\
&=&25-\dfrac{44}{3}+\dfrac{27}{4}+\left(10\right)\sqrt{5}+\left(\dfrac{35}{4}\right)\sqrt{5}+\dfrac{61}{5}-\left(\left(\dfrac{35}{4}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{5}\right)-\dfrac{135}{2}+\left(\dfrac{325}{9}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{32}{9}\right)\sqrt{5}+\left(\left(-\dfrac{124}{9}\right)\sqrt{5}\right)-\left(\left(-9\right)\sqrt{5}\right)+\left(\dfrac{37}{4}\right)\sqrt{5}\\
&=&-\dfrac{12523}{360}+\left(\dfrac{553}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(5\right)\sqrt{25}-\dfrac{44}{3}+\dfrac{27}{4}+\left(5\right)\sqrt{20}+\left(\dfrac{7}{4}\right)\sqrt{125}+\dfrac{61}{5}-\left(\left(\dfrac{7}{4}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{20}\right)+\left(-\dfrac{27}{2}\right)\sqrt{25}\right)-\left(\left(\dfrac{65}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{125}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{16}{9}\right)\sqrt{20}+\left(\left(-\dfrac{62}{9}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{45}\right)+\left(\dfrac{37}{8}\right)\sqrt{20}\right)\\
&=&\left(25-\dfrac{44}{3}+\dfrac{27}{4}+\left(10\right)\sqrt{5}+\left(\dfrac{35}{4}\right)\sqrt{5}+\dfrac{61}{5}-\left(\left(\dfrac{35}{4}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{5}\right)-\dfrac{135}{2}\right)-\left(\left(\dfrac{325}{9}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{32}{9}\right)\sqrt{5}+\left(\left(-\dfrac{124}{9}\right)\sqrt{5}\right)-\left(\left(-9\right)\sqrt{5}\right)+\left(\dfrac{37}{4}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{2293}{60}+\left(-\dfrac{35}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{973}{36}\right)\sqrt{5}+\dfrac{247}{72}\right)\\
&=&-\dfrac{2293}{60}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(-\dfrac{973}{36}\right)\sqrt{5}-\dfrac{247}{72}\\
&=&-\dfrac{14993}{360}+\left(-\dfrac{1393}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(5\right)\sqrt{25}-\dfrac{44}{3}+\dfrac{27}{4}+\left(5\right)\sqrt{20}+\left(\dfrac{7}{4}\right)\sqrt{125}+\dfrac{61}{5}-\left(\left(\dfrac{7}{4}\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{45}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{20}\right)+\left(-\dfrac{27}{2}\right)\sqrt{25}\right)\times\left(\left(\dfrac{65}{9}\right)\sqrt{125}+\left(-2\right)\sqrt{125}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{16}{9}\right)\sqrt{20}+\left(\left(-\dfrac{62}{9}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{45}\right)+\left(\dfrac{37}{8}\right)\sqrt{20}\right)\\
&=&\left(25-\dfrac{44}{3}+\dfrac{27}{4}+\left(10\right)\sqrt{5}+\left(\dfrac{35}{4}\right)\sqrt{5}+\dfrac{61}{5}-\left(\left(\dfrac{35}{4}\right)\sqrt{5}\right)-\left(\left(6\right)\sqrt{5}\right)-\left(\left(\dfrac{47}{3}\right)\sqrt{5}\right)-\dfrac{135}{2}\right)\times\left(\left(\dfrac{325}{9}\right)\sqrt{5}+\left(-10\right)\sqrt{5}+\dfrac{47}{8}-\dfrac{22}{9}+\left(-\dfrac{32}{9}\right)\sqrt{5}+\left(\left(-\dfrac{124}{9}\right)\sqrt{5}\right)-\left(\left(-9\right)\sqrt{5}\right)+\left(\dfrac{37}{4}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{2293}{60}+\left(-\dfrac{35}{3}\right)\sqrt{5}\right)\left(\left(\dfrac{973}{36}\right)\sqrt{5}+\dfrac{247}{72}\right)\\
&=&\left(-\dfrac{772513}{720}\right)\sqrt{5}-\dfrac{566371}{4320}+\left(-\dfrac{34055}{108}\right)\sqrt{25}\\
&=&\left(-\dfrac{772513}{720}\right)\sqrt{5}-\dfrac{7377371}{4320}\\
\end{eqnarray*}