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=\dfrac{33}{2}+2+\left(-\dfrac{20}{3}\right)\sqrt{20}+\left(\left(\dfrac{43}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{20}\right)+\left(\left(-\dfrac{77}{5}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)$ et $ Y=\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)-2-\left(\left(\left(\dfrac{43}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{125}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{20}+\left(9\right)\sqrt{20}+\left(-\dfrac{75}{2}\right)\sqrt{25}\right)$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\dfrac{33}{2}+2+\left(-\dfrac{20}{3}\right)\sqrt{20}+\left(\left(\dfrac{43}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{20}\right)+\left(\left(-\dfrac{77}{5}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)\right)+\left(\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)-2-\left(\left(\left(\dfrac{43}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{125}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{20}+\left(9\right)\sqrt{20}+\left(-\dfrac{75}{2}\right)\sqrt{25}\right)\right)\\ &=&\left(\dfrac{33}{2}+2+\left(-\dfrac{40}{3}\right)\sqrt{5}+\left(\left(\dfrac{129}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{87}{4}\right)\sqrt{5}\right)-\left(\left(-47\right)\sqrt{5}\right)+\left(\left(-\dfrac{154}{5}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)\right)+\left(\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)-2-\left(\left(\left(\dfrac{43}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{93}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{340}{9}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{5}+\left(18\right)\sqrt{5}-\dfrac{375}{2}\right)\right)\\ &=&\dfrac{33}{2}+2+\left(-\dfrac{40}{3}\right)\sqrt{5}+\left(\left(\dfrac{129}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{87}{4}\right)\sqrt{5}\right)-\left(\left(-47\right)\sqrt{5}\right)+\left(\left(-\dfrac{154}{5}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)+\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)-2-\left(\left(\left(\dfrac{43}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{93}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{340}{9}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{5}+\left(18\right)\sqrt{5}-\dfrac{375}{2}\right)\\ &=&204+\left(-\dfrac{5573}{180}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\dfrac{33}{2}+2+\left(-\dfrac{20}{3}\right)\sqrt{20}+\left(\left(\dfrac{43}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{20}\right)+\left(\left(-\dfrac{77}{5}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)-2-\left(\left(\left(\dfrac{43}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{125}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{20}+\left(9\right)\sqrt{20}+\left(-\dfrac{75}{2}\right)\sqrt{25}\right)\right)\\ &=&\left(\dfrac{33}{2}+2+\left(-\dfrac{40}{3}\right)\sqrt{5}+\left(\left(\dfrac{129}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{87}{4}\right)\sqrt{5}\right)-\left(\left(-47\right)\sqrt{5}\right)+\left(\left(-\dfrac{154}{5}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)-2-\left(\left(\left(\dfrac{43}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{93}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{340}{9}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{5}+\left(18\right)\sqrt{5}-\dfrac{375}{2}\right)\right)\\ &=&\left(\dfrac{37}{2}+\left(\dfrac{5437}{60}\right)\sqrt{5}\right)-\left(\left(-\dfrac{5471}{45}\right)\sqrt{5}+\dfrac{371}{2}\right)\\ &=&\dfrac{37}{2}+\left(\dfrac{5437}{60}\right)\sqrt{5}+\left(\dfrac{5471}{45}\right)\sqrt{5}-\dfrac{371}{2}\\ &=&-167+\left(\dfrac{7639}{36}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\dfrac{33}{2}+2+\left(-\dfrac{20}{3}\right)\sqrt{20}+\left(\left(\dfrac{43}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{47}{2}\right)\sqrt{20}\right)+\left(\left(-\dfrac{77}{5}\right)\sqrt{20}\right)-\left(\left(-9\right)\sqrt{125}\right)\right)\times\left(\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)-2-\left(\left(\left(\dfrac{43}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{125}\right)-\left(\left(5\right)\sqrt{45}\right)-\left(\left(-\dfrac{38}{5}\right)\sqrt{45}\right)\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{20}+\left(9\right)\sqrt{20}+\left(-\dfrac{75}{2}\right)\sqrt{25}\right)\right)\\ &=&\left(\dfrac{33}{2}+2+\left(-\dfrac{40}{3}\right)\sqrt{5}+\left(\left(\dfrac{129}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{87}{4}\right)\sqrt{5}\right)-\left(\left(-47\right)\sqrt{5}\right)+\left(\left(-\dfrac{154}{5}\right)\sqrt{5}\right)-\left(\left(-45\right)\sqrt{5}\right)\right)\times\left(\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)-2-\left(\left(\left(\dfrac{43}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{93}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{340}{9}\right)\sqrt{5}\right)-\left(\left(15\right)\sqrt{5}\right)-\left(\left(-\dfrac{114}{5}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{5}+\left(18\right)\sqrt{5}-\dfrac{375}{2}\right)\right)\\ &=&\left(\dfrac{37}{2}+\left(\dfrac{5437}{60}\right)\sqrt{5}\right)\left(\left(-\dfrac{5471}{45}\right)\sqrt{5}+\dfrac{371}{2}\right)\\ &=&\left(\dfrac{5241673}{360}\right)\sqrt{5}+\dfrac{13727}{4}+\left(-\dfrac{29745827}{2700}\right)\sqrt{25}\\ &=&\left(\dfrac{5241673}{360}\right)\sqrt{5}-\dfrac{13946341}{270}\\ \end{eqnarray*}