Ataraxy

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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.

Exercice

Soit $ X=\left(\dfrac{10}{3}\right)\sqrt{20}$ et $ Y=\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{125}\right)-\left(\left(-2\right)\sqrt{45}\right)-\left(\left(-7\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{45}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{1}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-7\right)\sqrt{125}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}+\dfrac{17}{4}+\left(\dfrac{47}{5}\right)\sqrt{125}\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(\dfrac{10}{3}\right)\sqrt{20}\right)+\left(\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{125}\right)-\left(\left(-2\right)\sqrt{45}\right)-\left(\left(-7\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{45}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{1}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-7\right)\sqrt{125}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}+\dfrac{17}{4}+\left(\dfrac{47}{5}\right)\sqrt{125}\right)\right)\\ &=&\left(\left(\dfrac{20}{3}\right)\sqrt{5}\right)+\left(\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{5}\right)-\left(\left(45\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{5}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}+\dfrac{17}{4}+\left(47\right)\sqrt{5}\right)\right)\\ &=&\left(\dfrac{20}{3}\right)\sqrt{5}+\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{5}\right)-\left(\left(45\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{5}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}+\dfrac{17}{4}+\left(47\right)\sqrt{5}\right)\\ &=&\left(-\dfrac{1051}{18}\right)\sqrt{5}+\dfrac{154}{15}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\dfrac{10}{3}\right)\sqrt{20}\right)-\left(\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{125}\right)-\left(\left(-2\right)\sqrt{45}\right)-\left(\left(-7\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{45}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{1}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-7\right)\sqrt{125}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}+\dfrac{17}{4}+\left(\dfrac{47}{5}\right)\sqrt{125}\right)\right)\\ &=&\left(\left(\dfrac{20}{3}\right)\sqrt{5}\right)-\left(\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{5}\right)-\left(\left(45\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{5}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}+\dfrac{17}{4}+\left(47\right)\sqrt{5}\right)\right)\\ &=&\left(\left(\dfrac{20}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1171}{18}\right)\sqrt{5}+\dfrac{154}{15}\right)\\ &=&\left(\dfrac{20}{3}\right)\sqrt{5}+\left(\dfrac{1171}{18}\right)\sqrt{5}-\dfrac{154}{15}\\ &=&\left(\dfrac{1291}{18}\right)\sqrt{5}-\dfrac{154}{15}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\dfrac{10}{3}\right)\sqrt{20}\right)\times\left(\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{45}\right)-\left(\left(9\right)\sqrt{125}\right)-\left(\left(-2\right)\sqrt{45}\right)-\left(\left(-7\right)\sqrt{45}\right)\right)-\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{45}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{1}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-7\right)\sqrt{125}\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{125}+\dfrac{17}{4}+\left(\dfrac{47}{5}\right)\sqrt{125}\right)\right)\\ &=&\left(\left(\dfrac{20}{3}\right)\sqrt{5}\right)\times\left(\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{5}\right)-\left(\left(45\right)\sqrt{5}\right)-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(-21\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)+\dfrac{77}{5}-\dfrac{69}{4}-\left(\left(-\dfrac{5}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{38}{3}-\left(\left(-35\right)\sqrt{5}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{200}{3}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{215}{9}\right)\sqrt{5}+\dfrac{17}{4}+\left(47\right)\sqrt{5}\right)\right)\\ &=&\left(\left(\dfrac{20}{3}\right)\sqrt{5}\right)\left(\left(-\dfrac{1171}{18}\right)\sqrt{5}+\dfrac{154}{15}\right)\\ &=&\left(-\dfrac{11710}{27}\right)\sqrt{25}+\left(\dfrac{616}{9}\right)\sqrt{5}\\ &=&-\dfrac{58550}{27}+\left(\dfrac{616}{9}\right)\sqrt{5}\\ \end{eqnarray*}