L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{12}{7}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{18}{5}\right)\sqrt{75}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\) et \( Y=\left(\left(6\right)\sqrt{12}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{74}{3}\right)\sqrt{27}+\left(-\dfrac{28}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{58}{9}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{12}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{12}{7}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{18}{5}\right)\sqrt{75}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\right)+\left(\left(\left(6\right)\sqrt{12}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{74}{3}\right)\sqrt{27}+\left(-\dfrac{28}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{58}{9}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{24}{7}\right)\sqrt{3}\right)-\left(\left(\left(-18\right)\sqrt{3}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\right)+\left(\left(\left(12\right)\sqrt{3}\right)-\left(\left(-5\right)\sqrt{3}+\left(74\right)\sqrt{3}+\left(-28\right)\sqrt{3}\right)-\dfrac{58}{3}-\left(\left(\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{26}{3}\right)\sqrt{3}\right)-\left(\left(-33\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{24}{7}\right)\sqrt{3}\right)-\left(\left(\left(-18\right)\sqrt{3}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)+\left(\left(12\right)\sqrt{3}\right)-\left(\left(-5\right)\sqrt{3}+\left(74\right)\sqrt{3}+\left(-28\right)\sqrt{3}\right)-\dfrac{58}{3}-\left(\left(\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{26}{3}\right)\sqrt{3}\right)-\left(\left(-33\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{458}{21}\right)\sqrt{3}+\dfrac{52}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{12}{7}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{18}{5}\right)\sqrt{75}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\right)-\left(\left(\left(6\right)\sqrt{12}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{74}{3}\right)\sqrt{27}+\left(-\dfrac{28}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{58}{9}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{24}{7}\right)\sqrt{3}\right)-\left(\left(\left(-18\right)\sqrt{3}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\right)-\left(\left(\left(12\right)\sqrt{3}\right)-\left(\left(-5\right)\sqrt{3}+\left(74\right)\sqrt{3}+\left(-28\right)\sqrt{3}\right)-\dfrac{58}{3}-\left(\left(\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{26}{3}\right)\sqrt{3}\right)-\left(\left(-33\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(\dfrac{102}{7}\right)\sqrt{3}+\dfrac{458}{21}\right)-\left(\left(-\dfrac{764}{21}\right)\sqrt{3}-\dfrac{58}{3}\right)\\
&=&\left(\dfrac{102}{7}\right)\sqrt{3}+\dfrac{458}{21}+\left(\dfrac{764}{21}\right)\sqrt{3}+\dfrac{58}{3}\\
&=&\left(\dfrac{1070}{21}\right)\sqrt{3}+\dfrac{288}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{12}{7}\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{18}{5}\right)\sqrt{75}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\right)\times\left(\left(\left(6\right)\sqrt{12}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{74}{3}\right)\sqrt{27}+\left(-\dfrac{28}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{58}{9}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{48}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{24}{7}\right)\sqrt{3}\right)-\left(\left(\left(-18\right)\sqrt{3}\right)-\dfrac{38}{3}-\dfrac{64}{7}\right)\right)\times\left(\left(\left(12\right)\sqrt{3}\right)-\left(\left(-5\right)\sqrt{3}+\left(74\right)\sqrt{3}+\left(-28\right)\sqrt{3}\right)-\dfrac{58}{3}-\left(\left(\left(-\dfrac{240}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{26}{3}\right)\sqrt{3}\right)-\left(\left(-33\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(\dfrac{102}{7}\right)\sqrt{3}+\dfrac{458}{21}\right)\left(\left(-\dfrac{764}{21}\right)\sqrt{3}-\dfrac{58}{3}\right)\\
&=&\left(-\dfrac{25976}{49}\right)\sqrt{9}+\left(-\dfrac{474148}{441}\right)\sqrt{3}-\dfrac{26564}{63}\\
&=&-\dfrac{887300}{441}+\left(-\dfrac{474148}{441}\right)\sqrt{3}\\
\end{eqnarray*}