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 \( X=\dfrac{2}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\left(-6\right)\sqrt{27}\right)-7+\left(-\dfrac{68}{5}\right)\sqrt{12}+\left(-\dfrac{67}{3}\right)\sqrt{12}+\left(-\dfrac{5}{3}\right)\sqrt{75}\) et \( Y=0+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(\dfrac{41}{2}\right)\sqrt{75}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{2}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\left(-6\right)\sqrt{27}\right)-7+\left(-\dfrac{68}{5}\right)\sqrt{12}+\left(-\dfrac{67}{3}\right)\sqrt{12}+\left(-\dfrac{5}{3}\right)\sqrt{75}\right)+\left(0+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(\dfrac{41}{2}\right)\sqrt{75}\right)\\
&=&\left(\dfrac{2}{5}-\dfrac{243}{5}-\left(\left(35\right)\sqrt{3}\right)+\left(\left(-18\right)\sqrt{3}\right)-7+\left(-\dfrac{136}{5}\right)\sqrt{3}+\left(-\dfrac{134}{3}\right)\sqrt{3}+\left(-\dfrac{25}{3}\right)\sqrt{3}\right)+\left(0+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(\dfrac{205}{2}\right)\sqrt{3}\right)\\
&=&\dfrac{2}{5}-\dfrac{243}{5}-\left(\left(35\right)\sqrt{3}\right)+\left(\left(-18\right)\sqrt{3}\right)-7+\left(-\dfrac{136}{5}\right)\sqrt{3}+\left(-\dfrac{134}{3}\right)\sqrt{3}+\left(-\dfrac{25}{3}\right)\sqrt{3}+0+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(\dfrac{205}{2}\right)\sqrt{3}\\
&=&-\dfrac{276}{5}+\left(-\dfrac{571}{10}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{2}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\left(-6\right)\sqrt{27}\right)-7+\left(-\dfrac{68}{5}\right)\sqrt{12}+\left(-\dfrac{67}{3}\right)\sqrt{12}+\left(-\dfrac{5}{3}\right)\sqrt{75}\right)-\left(0+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(\dfrac{41}{2}\right)\sqrt{75}\right)\\
&=&\left(\dfrac{2}{5}-\dfrac{243}{5}-\left(\left(35\right)\sqrt{3}\right)+\left(\left(-18\right)\sqrt{3}\right)-7+\left(-\dfrac{136}{5}\right)\sqrt{3}+\left(-\dfrac{134}{3}\right)\sqrt{3}+\left(-\dfrac{25}{3}\right)\sqrt{3}\right)-\left(0+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(\dfrac{205}{2}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{276}{5}+\left(-\dfrac{666}{5}\right)\sqrt{3}\right)-\left(0+\left(\dfrac{761}{10}\right)\sqrt{3}\right)\\
&=&-\dfrac{276}{5}+\left(-\dfrac{666}{5}\right)\sqrt{3}+0+\left(-\dfrac{761}{10}\right)\sqrt{3}\\
&=&-\dfrac{276}{5}+\left(-\dfrac{2093}{10}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{2}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{9}\right)-\left(\left(7\right)\sqrt{75}\right)+\left(\left(-6\right)\sqrt{27}\right)-7+\left(-\dfrac{68}{5}\right)\sqrt{12}+\left(-\dfrac{67}{3}\right)\sqrt{12}+\left(-\dfrac{5}{3}\right)\sqrt{75}\right)\times\left(0+\left(-\dfrac{66}{5}\right)\sqrt{12}+\left(\dfrac{41}{2}\right)\sqrt{75}\right)\\
&=&\left(\dfrac{2}{5}-\dfrac{243}{5}-\left(\left(35\right)\sqrt{3}\right)+\left(\left(-18\right)\sqrt{3}\right)-7+\left(-\dfrac{136}{5}\right)\sqrt{3}+\left(-\dfrac{134}{3}\right)\sqrt{3}+\left(-\dfrac{25}{3}\right)\sqrt{3}\right)\times\left(0+\left(-\dfrac{132}{5}\right)\sqrt{3}+\left(\dfrac{205}{2}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{276}{5}+\left(-\dfrac{666}{5}\right)\sqrt{3}\right)\left(0+\left(\dfrac{761}{10}\right)\sqrt{3}\right)\\
&=&0+\left(-\dfrac{105018}{25}\right)\sqrt{3}+\left(-\dfrac{253413}{25}\right)\sqrt{9}\\
&=&-\dfrac{760239}{25}+\left(-\dfrac{105018}{25}\right)\sqrt{3}\\
\end{eqnarray*}