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 \( X=\left(-\dfrac{21}{2}\right)\sqrt{75}\) et \( Y=\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{53}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)+\left(\dfrac{8}{9}\right)\sqrt{27}-\dfrac{15}{2}+\left(-\dfrac{8}{7}\right)\sqrt{9}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{43}{9}\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{21}{2}\right)\sqrt{75}\right)+\left(\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{53}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)+\left(\dfrac{8}{9}\right)\sqrt{27}-\dfrac{15}{2}+\left(-\dfrac{8}{7}\right)\sqrt{9}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{43}{9}\right)\sqrt{9}\right)\\
&=&\left(\left(-\dfrac{105}{2}\right)\sqrt{3}\right)+\left(\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\dfrac{69}{2}-\left(\left(-\dfrac{6}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{265}{2}\right)\sqrt{3}\right)-\left(\left(-11\right)\sqrt{3}\right)+\dfrac{51}{4}+\left(\dfrac{8}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{24}{7}+\left(\dfrac{108}{5}\right)\sqrt{3}-\dfrac{43}{3}\right)\\
&=&\left(-\dfrac{105}{2}\right)\sqrt{3}+\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\dfrac{69}{2}-\left(\left(-\dfrac{6}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{265}{2}\right)\sqrt{3}\right)-\left(\left(-11\right)\sqrt{3}\right)+\dfrac{51}{4}+\left(\dfrac{8}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{24}{7}+\left(\dfrac{108}{5}\right)\sqrt{3}-\dfrac{43}{3}\\
&=&\left(-\dfrac{31553}{420}\right)\sqrt{3}+\dfrac{1847}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{21}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{53}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)+\left(\dfrac{8}{9}\right)\sqrt{27}-\dfrac{15}{2}+\left(-\dfrac{8}{7}\right)\sqrt{9}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{43}{9}\right)\sqrt{9}\right)\\
&=&\left(\left(-\dfrac{105}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\dfrac{69}{2}-\left(\left(-\dfrac{6}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{265}{2}\right)\sqrt{3}\right)-\left(\left(-11\right)\sqrt{3}\right)+\dfrac{51}{4}+\left(\dfrac{8}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{24}{7}+\left(\dfrac{108}{5}\right)\sqrt{3}-\dfrac{43}{3}\right)\\
&=&\left(\left(-\dfrac{105}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{9503}{420}\right)\sqrt{3}+\dfrac{1847}{84}\right)\\
&=&\left(-\dfrac{105}{2}\right)\sqrt{3}+\left(\dfrac{9503}{420}\right)\sqrt{3}-\dfrac{1847}{84}\\
&=&\left(-\dfrac{12547}{420}\right)\sqrt{3}-\dfrac{1847}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{21}{2}\right)\sqrt{75}\right)\times\left(\left(\left(\dfrac{59}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{23}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{53}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{27}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{9}\right)+\left(\dfrac{8}{9}\right)\sqrt{27}-\dfrac{15}{2}+\left(-\dfrac{8}{7}\right)\sqrt{9}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{43}{9}\right)\sqrt{9}\right)\\
&=&\left(\left(-\dfrac{105}{2}\right)\sqrt{3}\right)\times\left(\left(\left(\dfrac{295}{4}\right)\sqrt{3}\right)+\dfrac{69}{2}-\left(\left(-\dfrac{6}{7}\right)\sqrt{3}\right)+\left(\left(-\dfrac{265}{2}\right)\sqrt{3}\right)-\left(\left(-11\right)\sqrt{3}\right)+\dfrac{51}{4}+\left(\dfrac{8}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{24}{7}+\left(\dfrac{108}{5}\right)\sqrt{3}-\dfrac{43}{3}\right)\\
&=&\left(\left(-\dfrac{105}{2}\right)\sqrt{3}\right)\left(\left(-\dfrac{9503}{420}\right)\sqrt{3}+\dfrac{1847}{84}\right)\\
&=&\left(\dfrac{9503}{8}\right)\sqrt{9}+\left(-\dfrac{9235}{8}\right)\sqrt{3}\\
&=&\dfrac{28509}{8}+\left(-\dfrac{9235}{8}\right)\sqrt{3}\\
\end{eqnarray*}