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 \( X=3-\dfrac{1}{3}+\left(-9\right)\sqrt{25}+\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{125}\right)\) et \( Y=\left(\dfrac{40}{3}\right)\sqrt{20}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(3-\dfrac{1}{3}+\left(-9\right)\sqrt{25}+\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{125}\right)\right)+\left(\left(\dfrac{40}{3}\right)\sqrt{20}\right)\\
&=&\left(3-\dfrac{1}{3}-45+\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{5}\right)\right)+\left(\left(\dfrac{80}{3}\right)\sqrt{5}\right)\\
&=&3-\dfrac{1}{3}-45+\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{5}\right)+\left(\dfrac{80}{3}\right)\sqrt{5}\\
&=&-\dfrac{127}{3}+\left(\dfrac{785}{42}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(3-\dfrac{1}{3}+\left(-9\right)\sqrt{25}+\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{125}\right)\right)-\left(\left(\dfrac{40}{3}\right)\sqrt{20}\right)\\
&=&\left(3-\dfrac{1}{3}-45+\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{80}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{127}{3}+\left(-\dfrac{335}{42}\right)\sqrt{5}\right)-\left(\left(\dfrac{80}{3}\right)\sqrt{5}\right)\\
&=&-\dfrac{127}{3}+\left(-\dfrac{335}{42}\right)\sqrt{5}+\left(-\dfrac{80}{3}\right)\sqrt{5}\\
&=&-\dfrac{127}{3}+\left(-\dfrac{485}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(3-\dfrac{1}{3}+\left(-9\right)\sqrt{25}+\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{125}\right)\right)\times\left(\left(\dfrac{40}{3}\right)\sqrt{20}\right)\\
&=&\left(3-\dfrac{1}{3}-45+\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{5}\right)\right)\times\left(\left(\dfrac{80}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{127}{3}+\left(-\dfrac{335}{42}\right)\sqrt{5}\right)\left(\left(\dfrac{80}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{10160}{9}\right)\sqrt{5}+\left(-\dfrac{13400}{63}\right)\sqrt{25}\\
&=&\left(-\dfrac{10160}{9}\right)\sqrt{5}-\dfrac{67000}{63}\\
\end{eqnarray*}