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 \( X=\left(\left(\dfrac{60}{7}\right)\sqrt{75}\right)+\dfrac{71}{9}-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(-3\right)\sqrt{12}\right)+\left(\left(-\dfrac{74}{7}\right)\sqrt{27}\right)-\left(\left(-8\right)\sqrt{12}\right)-\left(\left(-\dfrac{29}{5}\right)\sqrt{9}\right)+\dfrac{17}{4}\) et \( Y=\left(-\dfrac{70}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{12}+\left(-4\right)\sqrt{27}+\left(-\dfrac{69}{7}\right)\sqrt{12}+\left(-7\right)\sqrt{9}\) . 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{60}{7}\right)\sqrt{75}\right)+\dfrac{71}{9}-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(-3\right)\sqrt{12}\right)+\left(\left(-\dfrac{74}{7}\right)\sqrt{27}\right)-\left(\left(-8\right)\sqrt{12}\right)-\left(\left(-\dfrac{29}{5}\right)\sqrt{9}\right)+\dfrac{17}{4}\right)+\left(\left(-\dfrac{70}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{12}+\left(-4\right)\sqrt{27}+\left(-\dfrac{69}{7}\right)\sqrt{12}+\left(-7\right)\sqrt{9}\right)\\
&=&\left(\left(\left(\dfrac{300}{7}\right)\sqrt{3}\right)+\dfrac{71}{9}+18-\left(\left(-6\right)\sqrt{3}\right)+\left(\left(-\dfrac{222}{7}\right)\sqrt{3}\right)-\left(\left(-16\right)\sqrt{3}\right)+\dfrac{87}{5}+\dfrac{17}{4}\right)+\left(\left(-70\right)\sqrt{3}+\left(\dfrac{38}{3}\right)\sqrt{3}+\left(-12\right)\sqrt{3}+\left(-\dfrac{138}{7}\right)\sqrt{3}-21\right)\\
&=&\left(\left(\dfrac{300}{7}\right)\sqrt{3}\right)+\dfrac{71}{9}+18-\left(\left(-6\right)\sqrt{3}\right)+\left(\left(-\dfrac{222}{7}\right)\sqrt{3}\right)-\left(\left(-16\right)\sqrt{3}\right)+\dfrac{87}{5}+\dfrac{17}{4}+\left(-70\right)\sqrt{3}+\left(\dfrac{38}{3}\right)\sqrt{3}+\left(-12\right)\sqrt{3}+\left(-\dfrac{138}{7}\right)\sqrt{3}-21\\
&=&\left(-\dfrac{1174}{21}\right)\sqrt{3}+\dfrac{4777}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{60}{7}\right)\sqrt{75}\right)+\dfrac{71}{9}-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(-3\right)\sqrt{12}\right)+\left(\left(-\dfrac{74}{7}\right)\sqrt{27}\right)-\left(\left(-8\right)\sqrt{12}\right)-\left(\left(-\dfrac{29}{5}\right)\sqrt{9}\right)+\dfrac{17}{4}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{12}+\left(-4\right)\sqrt{27}+\left(-\dfrac{69}{7}\right)\sqrt{12}+\left(-7\right)\sqrt{9}\right)\\
&=&\left(\left(\left(\dfrac{300}{7}\right)\sqrt{3}\right)+\dfrac{71}{9}+18-\left(\left(-6\right)\sqrt{3}\right)+\left(\left(-\dfrac{222}{7}\right)\sqrt{3}\right)-\left(\left(-16\right)\sqrt{3}\right)+\dfrac{87}{5}+\dfrac{17}{4}\right)-\left(\left(-70\right)\sqrt{3}+\left(\dfrac{38}{3}\right)\sqrt{3}+\left(-12\right)\sqrt{3}+\left(-\dfrac{138}{7}\right)\sqrt{3}-21\right)\\
&=&\left(\left(\dfrac{232}{7}\right)\sqrt{3}+\dfrac{8557}{180}\right)-\left(\left(-\dfrac{1870}{21}\right)\sqrt{3}-21\right)\\
&=&\left(\dfrac{232}{7}\right)\sqrt{3}+\dfrac{8557}{180}+\left(\dfrac{1870}{21}\right)\sqrt{3}+21\\
&=&\left(\dfrac{2566}{21}\right)\sqrt{3}+\dfrac{12337}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{60}{7}\right)\sqrt{75}\right)+\dfrac{71}{9}-\left(\left(-6\right)\sqrt{9}\right)-\left(\left(-3\right)\sqrt{12}\right)+\left(\left(-\dfrac{74}{7}\right)\sqrt{27}\right)-\left(\left(-8\right)\sqrt{12}\right)-\left(\left(-\dfrac{29}{5}\right)\sqrt{9}\right)+\dfrac{17}{4}\right)\times\left(\left(-\dfrac{70}{3}\right)\sqrt{27}+\left(\dfrac{19}{3}\right)\sqrt{12}+\left(-4\right)\sqrt{27}+\left(-\dfrac{69}{7}\right)\sqrt{12}+\left(-7\right)\sqrt{9}\right)\\
&=&\left(\left(\left(\dfrac{300}{7}\right)\sqrt{3}\right)+\dfrac{71}{9}+18-\left(\left(-6\right)\sqrt{3}\right)+\left(\left(-\dfrac{222}{7}\right)\sqrt{3}\right)-\left(\left(-16\right)\sqrt{3}\right)+\dfrac{87}{5}+\dfrac{17}{4}\right)\times\left(\left(-70\right)\sqrt{3}+\left(\dfrac{38}{3}\right)\sqrt{3}+\left(-12\right)\sqrt{3}+\left(-\dfrac{138}{7}\right)\sqrt{3}-21\right)\\
&=&\left(\left(\dfrac{232}{7}\right)\sqrt{3}+\dfrac{8557}{180}\right)\left(\left(-\dfrac{1870}{21}\right)\sqrt{3}-21\right)\\
&=&\left(-\dfrac{433840}{147}\right)\sqrt{9}+\left(-\dfrac{1863247}{378}\right)\sqrt{3}-\dfrac{59899}{60}\\
&=&-\dfrac{28965451}{2940}+\left(-\dfrac{1863247}{378}\right)\sqrt{3}\\
\end{eqnarray*}