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 \( X=\left(\dfrac{28}{3}\right)\sqrt{9}+\left(-\dfrac{64}{5}\right)\sqrt{27}+\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{75}\right)+\left(\left(-\dfrac{13}{9}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)\) et \( Y=\dfrac{1}{3}\) . 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{28}{3}\right)\sqrt{9}+\left(-\dfrac{64}{5}\right)\sqrt{27}+\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{75}\right)+\left(\left(-\dfrac{13}{9}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)\right)+\left(\dfrac{1}{3}\right)\\
&=&\left(28+\left(-\dfrac{192}{5}\right)\sqrt{3}+\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)+29-\left(\left(-\dfrac{200}{3}\right)\sqrt{3}\right)+\left(\left(-\dfrac{26}{9}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)\right)+\left(\dfrac{1}{3}\right)\\
&=&28+\left(-\dfrac{192}{5}\right)\sqrt{3}+\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)+29-\left(\left(-\dfrac{200}{3}\right)\sqrt{3}\right)+\left(\left(-\dfrac{26}{9}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)+\dfrac{1}{3}\\
&=&\dfrac{172}{3}+\left(\dfrac{1817}{45}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{28}{3}\right)\sqrt{9}+\left(-\dfrac{64}{5}\right)\sqrt{27}+\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{75}\right)+\left(\left(-\dfrac{13}{9}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)\right)-\left(\dfrac{1}{3}\right)\\
&=&\left(28+\left(-\dfrac{192}{5}\right)\sqrt{3}+\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)+29-\left(\left(-\dfrac{200}{3}\right)\sqrt{3}\right)+\left(\left(-\dfrac{26}{9}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)\right)-\left(\dfrac{1}{3}\right)\\
&=&\left(57+\left(\dfrac{1817}{45}\right)\sqrt{3}\right)-\left(\dfrac{1}{3}\right)\\
&=&57+\left(\dfrac{1817}{45}\right)\sqrt{3}+-\dfrac{1}{3}\\
&=&\dfrac{170}{3}+\left(\dfrac{1817}{45}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{28}{3}\right)\sqrt{9}+\left(-\dfrac{64}{5}\right)\sqrt{27}+\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{9}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{75}\right)+\left(\left(-\dfrac{13}{9}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{27}\right)-\left(\left(-\dfrac{9}{8}\right)\sqrt{75}\right)\right)\times\left(\dfrac{1}{3}\right)\\
&=&\left(28+\left(-\dfrac{192}{5}\right)\sqrt{3}+\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)+29-\left(\left(-\dfrac{200}{3}\right)\sqrt{3}\right)+\left(\left(-\dfrac{26}{9}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{3}\right)\right)\times\left(\dfrac{1}{3}\right)\\
&=&\left(57+\left(\dfrac{1817}{45}\right)\sqrt{3}\right)\left(\dfrac{1}{3}\right)\\
&=&19+\left(\dfrac{1817}{135}\right)\sqrt{3}\\
&=&19+\left(\dfrac{1817}{135}\right)\sqrt{3}\\
\end{eqnarray*}