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 \( X=\left(\dfrac{75}{4}\right)\sqrt{125}\) et \( Y=\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{52}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{79}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{75}{4}\right)\sqrt{125}\right)+\left(\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{52}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{79}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{5}\right)+\left(-45-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{52}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{395}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\dfrac{375}{4}\right)\sqrt{5}-45-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{52}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{395}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{4331}{36}\right)\sqrt{5}-45\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{75}{4}\right)\sqrt{125}\right)-\left(\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{52}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{79}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{5}\right)-\left(-45-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{52}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{395}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{5}\right)-\left(-45+\left(\dfrac{239}{9}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{375}{4}\right)\sqrt{5}+45+\left(-\dfrac{239}{9}\right)\sqrt{5}\\
&=&\left(\dfrac{2419}{36}\right)\sqrt{5}+45\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{75}{4}\right)\sqrt{125}\right)\times\left(\left(\left(-9\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)-\left(\left(\left(\dfrac{52}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{79}{9}\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{5}\right)\times\left(-45-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{52}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{395}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{375}{4}\right)\sqrt{5}\right)\left(-45+\left(\dfrac{239}{9}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{16875}{4}\right)\sqrt{5}+\left(\dfrac{29875}{12}\right)\sqrt{25}\\
&=&\left(-\dfrac{16875}{4}\right)\sqrt{5}+\dfrac{149375}{12}\\
\end{eqnarray*}