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 \( X=\left(-1-\left(\left(\dfrac{73}{3}\right)\sqrt{75}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{27}+\left(-\dfrac{68}{7}\right)\sqrt{9}\right)\) et \( Y=\left(\left(-\dfrac{61}{4}\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{27}+\left(-\dfrac{19}{2}\right)\sqrt{27}-\dfrac{78}{7}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-1-\left(\left(\dfrac{73}{3}\right)\sqrt{75}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{27}+\left(-\dfrac{68}{7}\right)\sqrt{9}\right)\right)+\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{27}+\left(-\dfrac{19}{2}\right)\sqrt{27}-\dfrac{78}{7}\right)\right)\\
&=&\left(\left(-1-\left(\left(\dfrac{365}{3}\right)\sqrt{3}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{3}-\dfrac{204}{7}\right)\right)+\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{33}{8}\right)\sqrt{3}+\left(-\dfrac{57}{2}\right)\sqrt{3}-\dfrac{78}{7}\right)\right)\\
&=&\left(-1-\left(\left(\dfrac{365}{3}\right)\sqrt{3}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{3}-\dfrac{204}{7}\right)+\left(\left(-\dfrac{305}{4}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{33}{8}\right)\sqrt{3}+\left(-\dfrac{57}{2}\right)\sqrt{3}-\dfrac{78}{7}\right)\\
&=&\dfrac{227}{7}+\left(-\dfrac{3931}{24}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-1-\left(\left(\dfrac{73}{3}\right)\sqrt{75}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{27}+\left(-\dfrac{68}{7}\right)\sqrt{9}\right)\right)-\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{27}+\left(-\dfrac{19}{2}\right)\sqrt{27}-\dfrac{78}{7}\right)\right)\\
&=&\left(\left(-1-\left(\left(\dfrac{365}{3}\right)\sqrt{3}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{3}-\dfrac{204}{7}\right)\right)-\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{33}{8}\right)\sqrt{3}+\left(-\dfrac{57}{2}\right)\sqrt{3}-\dfrac{78}{7}\right)\right)\\
&=&\left(\dfrac{149}{7}+\left(-\dfrac{1523}{12}\right)\sqrt{3}\right)-\left(\left(-\dfrac{295}{8}\right)\sqrt{3}+\dfrac{78}{7}\right)\\
&=&\dfrac{149}{7}+\left(-\dfrac{1523}{12}\right)\sqrt{3}+\left(\dfrac{295}{8}\right)\sqrt{3}-\dfrac{78}{7}\\
&=&\dfrac{71}{7}+\left(-\dfrac{2161}{24}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-1-\left(\left(\dfrac{73}{3}\right)\sqrt{75}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{7}{4}\right)\sqrt{27}+\left(-\dfrac{68}{7}\right)\sqrt{9}\right)\right)\times\left(\left(\left(-\dfrac{61}{4}\right)\sqrt{75}\right)-\left(\left(-3\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{27}+\left(-\dfrac{19}{2}\right)\sqrt{27}-\dfrac{78}{7}\right)\right)\\
&=&\left(\left(-1-\left(\left(\dfrac{365}{3}\right)\sqrt{3}\right)-\dfrac{48}{7}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{3}-\dfrac{204}{7}\right)\right)\times\left(\left(\left(-\dfrac{305}{4}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(\dfrac{33}{8}\right)\sqrt{3}+\left(-\dfrac{57}{2}\right)\sqrt{3}-\dfrac{78}{7}\right)\right)\\
&=&\left(\dfrac{149}{7}+\left(-\dfrac{1523}{12}\right)\sqrt{3}\right)\left(\left(-\dfrac{295}{8}\right)\sqrt{3}+\dfrac{78}{7}\right)\\
&=&\left(-\dfrac{17593}{8}\right)\sqrt{3}+\dfrac{11622}{49}+\left(\dfrac{449285}{96}\right)\sqrt{9}\\
&=&\left(-\dfrac{17593}{8}\right)\sqrt{3}+\dfrac{22386869}{1568}\\
\end{eqnarray*}