L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{11}{7}\right)\sqrt{49}-\dfrac{1}{3}+\left(\dfrac{61}{3}\right)\sqrt{49}+\left(\dfrac{36}{5}\right)\sqrt{63}+\left(-\dfrac{41}{4}\right)\sqrt{49}+\left(\dfrac{64}{9}\right)\sqrt{28}+\left(-\dfrac{20}{9}\right)\sqrt{28}+\left(\dfrac{81}{7}\right)\sqrt{49}+\left(\left(6\right)\sqrt{28}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{63}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{32}{7}\right)\sqrt{63}\right)\) et \( Y=\left(-\dfrac{3}{7}\right)\sqrt{63}\) . 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{11}{7}\right)\sqrt{49}-\dfrac{1}{3}+\left(\dfrac{61}{3}\right)\sqrt{49}+\left(\dfrac{36}{5}\right)\sqrt{63}+\left(-\dfrac{41}{4}\right)\sqrt{49}+\left(\dfrac{64}{9}\right)\sqrt{28}+\left(-\dfrac{20}{9}\right)\sqrt{28}+\left(\dfrac{81}{7}\right)\sqrt{49}+\left(\left(6\right)\sqrt{28}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{63}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{32}{7}\right)\sqrt{63}\right)\right)+\left(\left(-\dfrac{3}{7}\right)\sqrt{63}\right)\\
&=&\left(11-\dfrac{1}{3}+\dfrac{427}{3}+\left(\dfrac{108}{5}\right)\sqrt{7}-\dfrac{287}{4}+\left(\dfrac{128}{9}\right)\sqrt{7}+\left(-\dfrac{40}{9}\right)\sqrt{7}+81+\left(\left(12\right)\sqrt{7}\right)-\left(\left(-11\right)\sqrt{7}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{96}{7}\right)\sqrt{7}\right)\right)+\left(\left(-\dfrac{9}{7}\right)\sqrt{7}\right)\\
&=&11-\dfrac{1}{3}+\dfrac{427}{3}+\left(\dfrac{108}{5}\right)\sqrt{7}-\dfrac{287}{4}+\left(\dfrac{128}{9}\right)\sqrt{7}+\left(-\dfrac{40}{9}\right)\sqrt{7}+81+\left(\left(12\right)\sqrt{7}\right)-\left(\left(-11\right)\sqrt{7}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{96}{7}\right)\sqrt{7}\right)+\left(-\dfrac{9}{7}\right)\sqrt{7}\\
&=&\dfrac{3233}{20}+\left(\dfrac{21044}{315}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{11}{7}\right)\sqrt{49}-\dfrac{1}{3}+\left(\dfrac{61}{3}\right)\sqrt{49}+\left(\dfrac{36}{5}\right)\sqrt{63}+\left(-\dfrac{41}{4}\right)\sqrt{49}+\left(\dfrac{64}{9}\right)\sqrt{28}+\left(-\dfrac{20}{9}\right)\sqrt{28}+\left(\dfrac{81}{7}\right)\sqrt{49}+\left(\left(6\right)\sqrt{28}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{63}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{32}{7}\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{63}\right)\\
&=&\left(11-\dfrac{1}{3}+\dfrac{427}{3}+\left(\dfrac{108}{5}\right)\sqrt{7}-\dfrac{287}{4}+\left(\dfrac{128}{9}\right)\sqrt{7}+\left(-\dfrac{40}{9}\right)\sqrt{7}+81+\left(\left(12\right)\sqrt{7}\right)-\left(\left(-11\right)\sqrt{7}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{96}{7}\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{9}{7}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{3233}{20}+\left(\dfrac{21449}{315}\right)\sqrt{7}\right)-\left(\left(-\dfrac{9}{7}\right)\sqrt{7}\right)\\
&=&\dfrac{3233}{20}+\left(\dfrac{21449}{315}\right)\sqrt{7}+\left(\dfrac{9}{7}\right)\sqrt{7}\\
&=&\dfrac{3233}{20}+\left(\dfrac{3122}{45}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{11}{7}\right)\sqrt{49}-\dfrac{1}{3}+\left(\dfrac{61}{3}\right)\sqrt{49}+\left(\dfrac{36}{5}\right)\sqrt{63}+\left(-\dfrac{41}{4}\right)\sqrt{49}+\left(\dfrac{64}{9}\right)\sqrt{28}+\left(-\dfrac{20}{9}\right)\sqrt{28}+\left(\dfrac{81}{7}\right)\sqrt{49}+\left(\left(6\right)\sqrt{28}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{63}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{32}{7}\right)\sqrt{63}\right)\right)\times\left(\left(-\dfrac{3}{7}\right)\sqrt{63}\right)\\
&=&\left(11-\dfrac{1}{3}+\dfrac{427}{3}+\left(\dfrac{108}{5}\right)\sqrt{7}-\dfrac{287}{4}+\left(\dfrac{128}{9}\right)\sqrt{7}+\left(-\dfrac{40}{9}\right)\sqrt{7}+81+\left(\left(12\right)\sqrt{7}\right)-\left(\left(-11\right)\sqrt{7}\right)-\dfrac{3}{5}-\left(\left(-\dfrac{96}{7}\right)\sqrt{7}\right)\right)\times\left(\left(-\dfrac{9}{7}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{3233}{20}+\left(\dfrac{21449}{315}\right)\sqrt{7}\right)\left(\left(-\dfrac{9}{7}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{29097}{140}\right)\sqrt{7}+\left(-\dfrac{21449}{245}\right)\sqrt{49}\\
&=&\left(-\dfrac{29097}{140}\right)\sqrt{7}-\dfrac{21449}{35}\\
\end{eqnarray*}