L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{10}{9}\right)\sqrt{28}\) et \( Y=\left(\left(-\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{28}\right)+\dfrac{31}{2}+\left(\left(\dfrac{25}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{175}\right)+\left(-\dfrac{23}{3}\right)\sqrt{175}+\dfrac{7}{2}+\left(\dfrac{16}{9}\right)\sqrt{63}+\left(-2\right)\sqrt{63}\) . 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{10}{9}\right)\sqrt{28}\right)+\left(\left(\left(-\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{28}\right)+\dfrac{31}{2}+\left(\left(\dfrac{25}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{175}\right)+\left(-\dfrac{23}{3}\right)\sqrt{175}+\dfrac{7}{2}+\left(\dfrac{16}{9}\right)\sqrt{63}+\left(-2\right)\sqrt{63}\right)\\
&=&\left(\left(\dfrac{20}{9}\right)\sqrt{7}\right)+\left(\left(\left(-\dfrac{51}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{7}\right)+\dfrac{31}{2}+\dfrac{175}{3}-\dfrac{525}{2}-\left(\left(\dfrac{25}{2}\right)\sqrt{7}\right)+\left(-\dfrac{115}{3}\right)\sqrt{7}+\dfrac{7}{2}+\left(\dfrac{16}{3}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\\
&=&\left(\dfrac{20}{9}\right)\sqrt{7}+\left(\left(-\dfrac{51}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{7}\right)+\dfrac{31}{2}+\dfrac{175}{3}-\dfrac{525}{2}-\left(\left(\dfrac{25}{2}\right)\sqrt{7}\right)+\left(-\dfrac{115}{3}\right)\sqrt{7}+\dfrac{7}{2}+\left(\dfrac{16}{3}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\\
&=&\left(-\dfrac{2663}{45}\right)\sqrt{7}-\dfrac{1111}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{28}\right)-\left(\left(\left(-\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{28}\right)+\dfrac{31}{2}+\left(\left(\dfrac{25}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{175}\right)+\left(-\dfrac{23}{3}\right)\sqrt{175}+\dfrac{7}{2}+\left(\dfrac{16}{9}\right)\sqrt{63}+\left(-2\right)\sqrt{63}\right)\\
&=&\left(\left(\dfrac{20}{9}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{51}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{7}\right)+\dfrac{31}{2}+\dfrac{175}{3}-\dfrac{525}{2}-\left(\left(\dfrac{25}{2}\right)\sqrt{7}\right)+\left(-\dfrac{115}{3}\right)\sqrt{7}+\dfrac{7}{2}+\left(\dfrac{16}{3}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{20}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{307}{5}\right)\sqrt{7}-\dfrac{1111}{6}\right)\\
&=&\left(\dfrac{20}{9}\right)\sqrt{7}+\left(\dfrac{307}{5}\right)\sqrt{7}+\dfrac{1111}{6}\\
&=&\left(\dfrac{2863}{45}\right)\sqrt{7}+\dfrac{1111}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{10}{9}\right)\sqrt{28}\right)\times\left(\left(\left(-\dfrac{17}{2}\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{5}\right)\sqrt{28}\right)+\dfrac{31}{2}+\left(\left(\dfrac{25}{3}\right)\sqrt{49}\right)-\left(\left(\dfrac{75}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{175}\right)+\left(-\dfrac{23}{3}\right)\sqrt{175}+\dfrac{7}{2}+\left(\dfrac{16}{9}\right)\sqrt{63}+\left(-2\right)\sqrt{63}\right)\\
&=&\left(\left(\dfrac{20}{9}\right)\sqrt{7}\right)\times\left(\left(\left(-\dfrac{51}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{7}\right)+\dfrac{31}{2}+\dfrac{175}{3}-\dfrac{525}{2}-\left(\left(\dfrac{25}{2}\right)\sqrt{7}\right)+\left(-\dfrac{115}{3}\right)\sqrt{7}+\dfrac{7}{2}+\left(\dfrac{16}{3}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{20}{9}\right)\sqrt{7}\right)\left(\left(-\dfrac{307}{5}\right)\sqrt{7}-\dfrac{1111}{6}\right)\\
&=&\left(-\dfrac{1228}{9}\right)\sqrt{49}+\left(-\dfrac{11110}{27}\right)\sqrt{7}\\
&=&-\dfrac{8596}{9}+\left(-\dfrac{11110}{27}\right)\sqrt{7}\\
\end{eqnarray*}