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(\left(\left(-\dfrac{1}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{8}\right)\sqrt{12}+\left(\dfrac{47}{9}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{27}+\left(-\dfrac{29}{2}\right)\sqrt{75}+\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(-\dfrac{59}{9}\right)\sqrt{75}+\left(\dfrac{66}{7}\right)\sqrt{12}\right)$ et $ Y=\left(\dfrac{41}{7}\right)\sqrt{9}+\left(\dfrac{43}{4}\right)\sqrt{12}-4+\left(0\right)\sqrt{27}+\left(-5\right)\sqrt{27}+\left(\left(\dfrac{66}{7}\right)\sqrt{12}\right)-\dfrac{47}{7}-\left(\left(-9\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)+\left(-7\right)\sqrt{75}+\left(\left(-\dfrac{39}{2}\right)\sqrt{12}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-2\right)\sqrt{75}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{8}\right)\sqrt{12}+\left(\dfrac{47}{9}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{27}+\left(-\dfrac{29}{2}\right)\sqrt{75}+\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(-\dfrac{59}{9}\right)\sqrt{75}+\left(\dfrac{66}{7}\right)\sqrt{12}\right)\right)+\left(\left(\dfrac{41}{7}\right)\sqrt{9}+\left(\dfrac{43}{4}\right)\sqrt{12}-4+\left(0\right)\sqrt{27}+\left(-5\right)\sqrt{27}+\left(\left(\dfrac{66}{7}\right)\sqrt{12}\right)-\dfrac{47}{7}-\left(\left(-9\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)+\left(-7\right)\sqrt{75}+\left(\left(-\dfrac{39}{2}\right)\sqrt{12}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-2\right)\sqrt{75}\right)\\ &=&\left(\left(\left(\left(-1\right)\sqrt{3}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{4}\right)\sqrt{3}+\left(\dfrac{94}{9}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(-\dfrac{145}{2}\right)\sqrt{3}+\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(-\dfrac{295}{9}\right)\sqrt{3}+\left(\dfrac{132}{7}\right)\sqrt{3}\right)\right)+\left(\dfrac{123}{7}+\left(\dfrac{43}{2}\right)\sqrt{3}-4+\left(0\right)\sqrt{3}+\left(-15\right)\sqrt{3}+\left(\left(\dfrac{132}{7}\right)\sqrt{3}\right)-\dfrac{47}{7}-\left(\left(-18\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)+\left(-35\right)\sqrt{3}+\left(\left(-39\right)\sqrt{3}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-10\right)\sqrt{3}\right)\\ &=&\left(\left(\left(-1\right)\sqrt{3}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{4}\right)\sqrt{3}+\left(\dfrac{94}{9}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(-\dfrac{145}{2}\right)\sqrt{3}+\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(-\dfrac{295}{9}\right)\sqrt{3}+\left(\dfrac{132}{7}\right)\sqrt{3}\right)+\dfrac{123}{7}+\left(\dfrac{43}{2}\right)\sqrt{3}-4+\left(0\right)\sqrt{3}+\left(-15\right)\sqrt{3}+\left(\left(\dfrac{132}{7}\right)\sqrt{3}\right)-\dfrac{47}{7}-\left(\left(-18\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)+\left(-35\right)\sqrt{3}+\left(\left(-39\right)\sqrt{3}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-10\right)\sqrt{3}\\ &=&\left(\dfrac{769}{8}\right)\sqrt{3}-\dfrac{2063}{210}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{8}\right)\sqrt{12}+\left(\dfrac{47}{9}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{27}+\left(-\dfrac{29}{2}\right)\sqrt{75}+\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(-\dfrac{59}{9}\right)\sqrt{75}+\left(\dfrac{66}{7}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{41}{7}\right)\sqrt{9}+\left(\dfrac{43}{4}\right)\sqrt{12}-4+\left(0\right)\sqrt{27}+\left(-5\right)\sqrt{27}+\left(\left(\dfrac{66}{7}\right)\sqrt{12}\right)-\dfrac{47}{7}-\left(\left(-9\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)+\left(-7\right)\sqrt{75}+\left(\left(-\dfrac{39}{2}\right)\sqrt{12}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-2\right)\sqrt{75}\right)\\ &=&\left(\left(\left(\left(-1\right)\sqrt{3}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{4}\right)\sqrt{3}+\left(\dfrac{94}{9}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(-\dfrac{145}{2}\right)\sqrt{3}+\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(-\dfrac{295}{9}\right)\sqrt{3}+\left(\dfrac{132}{7}\right)\sqrt{3}\right)\right)-\left(\dfrac{123}{7}+\left(\dfrac{43}{2}\right)\sqrt{3}-4+\left(0\right)\sqrt{3}+\left(-15\right)\sqrt{3}+\left(\left(\dfrac{132}{7}\right)\sqrt{3}\right)-\dfrac{47}{7}-\left(\left(-18\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)+\left(-35\right)\sqrt{3}+\left(\left(-39\right)\sqrt{3}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-10\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{6875}{56}\right)\sqrt{3}-\dfrac{12}{7}\right)-\left(-\dfrac{1703}{210}+\left(-\dfrac{373}{14}\right)\sqrt{3}\right)\\ &=&\left(\dfrac{6875}{56}\right)\sqrt{3}-\dfrac{12}{7}+\dfrac{1703}{210}+\left(\dfrac{373}{14}\right)\sqrt{3}\\ &=&\left(\dfrac{8367}{56}\right)\sqrt{3}+\dfrac{1343}{210}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\left(-\dfrac{1}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{8}\right)\sqrt{12}+\left(\dfrac{47}{9}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{27}+\left(-\dfrac{29}{2}\right)\sqrt{75}+\left(-\dfrac{27}{8}\right)\sqrt{75}+\left(-\dfrac{59}{9}\right)\sqrt{75}+\left(\dfrac{66}{7}\right)\sqrt{12}\right)\right)\times\left(\left(\dfrac{41}{7}\right)\sqrt{9}+\left(\dfrac{43}{4}\right)\sqrt{12}-4+\left(0\right)\sqrt{27}+\left(-5\right)\sqrt{27}+\left(\left(\dfrac{66}{7}\right)\sqrt{12}\right)-\dfrac{47}{7}-\left(\left(-9\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)+\left(-7\right)\sqrt{75}+\left(\left(-\dfrac{39}{2}\right)\sqrt{12}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-2\right)\sqrt{75}\right)\\ &=&\left(\left(\left(\left(-1\right)\sqrt{3}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{12}{7}+\left(-\dfrac{33}{4}\right)\sqrt{3}+\left(\dfrac{94}{9}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}+\left(-\dfrac{145}{2}\right)\sqrt{3}+\left(-\dfrac{135}{8}\right)\sqrt{3}+\left(-\dfrac{295}{9}\right)\sqrt{3}+\left(\dfrac{132}{7}\right)\sqrt{3}\right)\right)\times\left(\dfrac{123}{7}+\left(\dfrac{43}{2}\right)\sqrt{3}-4+\left(0\right)\sqrt{3}+\left(-15\right)\sqrt{3}+\left(\left(\dfrac{132}{7}\right)\sqrt{3}\right)-\dfrac{47}{7}-\left(\left(-18\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)+\left(-35\right)\sqrt{3}+\left(\left(-39\right)\sqrt{3}\right)-\dfrac{64}{5}-\dfrac{13}{6}+\left(-10\right)\sqrt{3}\right)\\ &=&\left(\left(\dfrac{6875}{56}\right)\sqrt{3}-\dfrac{12}{7}\right)\left(-\dfrac{1703}{210}+\left(-\dfrac{373}{14}\right)\sqrt{3}\right)\\ &=&\left(-\dfrac{2234201}{2352}\right)\sqrt{3}+\left(-\dfrac{2564375}{784}\right)\sqrt{9}+\dfrac{3406}{245}\\ &=&\left(-\dfrac{2234201}{2352}\right)\sqrt{3}-\dfrac{38411129}{3920}\\ \end{eqnarray*}