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 \( X=\left(\left(-\dfrac{35}{8}\right)\sqrt{9}+0+\left(-\dfrac{45}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{79}{9}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(5\right)\sqrt{27}\right)\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{27}+\left(-2\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{27}\right)\) et \( Y=\left(\left(\left(-1\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{9}\right)\) . 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{35}{8}\right)\sqrt{9}+0+\left(-\dfrac{45}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{79}{9}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(5\right)\sqrt{27}\right)\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{27}+\left(-2\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{27}\right)\right)+\left(\left(\left(\left(-1\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(-\dfrac{105}{8}+0+\left(-\dfrac{135}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{3}+\left(-4\right)\sqrt{3}+\left(-70\right)\sqrt{3}\right)\right)+\left(\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{7}\right)\sqrt{3}\right)\right)-\dfrac{198}{7}\right)\\
&=&\left(-\dfrac{105}{8}+0+\left(-\dfrac{135}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{3}+\left(-4\right)\sqrt{3}+\left(-70\right)\sqrt{3}\right)+\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{7}\right)\sqrt{3}\right)\right)-\dfrac{198}{7}\\
&=&-\dfrac{2319}{56}+\left(\dfrac{7471}{252}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{35}{8}\right)\sqrt{9}+0+\left(-\dfrac{45}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{79}{9}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(5\right)\sqrt{27}\right)\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{27}+\left(-2\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{27}\right)\right)-\left(\left(\left(\left(-1\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(-\dfrac{105}{8}+0+\left(-\dfrac{135}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{3}+\left(-4\right)\sqrt{3}+\left(-70\right)\sqrt{3}\right)\right)-\left(\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{7}\right)\sqrt{3}\right)\right)-\dfrac{198}{7}\right)\\
&=&\left(-\dfrac{105}{8}+\left(\dfrac{11791}{252}\right)\sqrt{3}\right)-\left(\left(-\dfrac{120}{7}\right)\sqrt{3}-\dfrac{198}{7}\right)\\
&=&-\dfrac{105}{8}+\left(\dfrac{11791}{252}\right)\sqrt{3}+\left(\dfrac{120}{7}\right)\sqrt{3}+\dfrac{198}{7}\\
&=&\dfrac{849}{56}+\left(\dfrac{16111}{252}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{35}{8}\right)\sqrt{9}+0+\left(-\dfrac{45}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{79}{9}\right)\sqrt{12}\right)-\left(\left(\left(\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(5\right)\sqrt{27}\right)\right)-\left(\left(\dfrac{27}{7}\right)\sqrt{27}+\left(-2\right)\sqrt{12}+\left(-\dfrac{70}{3}\right)\sqrt{27}\right)\right)\times\left(\left(\left(\left(-1\right)\sqrt{75}\right)-\left(\left(\dfrac{17}{7}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{66}{7}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(-\dfrac{105}{8}+0+\left(-\dfrac{135}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\left(\left(15\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{3}+\left(-4\right)\sqrt{3}+\left(-70\right)\sqrt{3}\right)\right)\times\left(\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(\dfrac{85}{7}\right)\sqrt{3}\right)\right)-\dfrac{198}{7}\right)\\
&=&\left(-\dfrac{105}{8}+\left(\dfrac{11791}{252}\right)\sqrt{3}\right)\left(\left(-\dfrac{120}{7}\right)\sqrt{3}-\dfrac{198}{7}\right)\\
&=&\left(-\dfrac{107651}{98}\right)\sqrt{3}+\dfrac{1485}{4}+\left(-\dfrac{117910}{147}\right)\sqrt{9}\\
&=&\left(-\dfrac{107651}{98}\right)\sqrt{3}-\dfrac{398875}{196}\\
\end{eqnarray*}