L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(6\right)\sqrt{25}+\left(\left(-\dfrac{79}{3}\right)\sqrt{20}\right)-\left(\left(-6\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(-2\right)\sqrt{25}+\left(6\right)\sqrt{25}+\left(-\dfrac{62}{5}\right)\sqrt{125}\) et \( Y=-\dfrac{18}{7}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(6\right)\sqrt{25}+\left(\left(-\dfrac{79}{3}\right)\sqrt{20}\right)-\left(\left(-6\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(-2\right)\sqrt{25}+\left(6\right)\sqrt{25}+\left(-\dfrac{62}{5}\right)\sqrt{125}\right)+\left(-\dfrac{18}{7}\right)\\
&=&\left(30+\left(\left(-\dfrac{158}{3}\right)\sqrt{5}\right)-\left(\left(-30\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)-10+30+\left(-62\right)\sqrt{5}\right)+\left(-\dfrac{18}{7}\right)\\
&=&30+\left(\left(-\dfrac{158}{3}\right)\sqrt{5}\right)-\left(\left(-30\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)-10+30+\left(-62\right)\sqrt{5}-\dfrac{18}{7}\\
&=&\dfrac{332}{7}+\left(-\dfrac{343}{6}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(6\right)\sqrt{25}+\left(\left(-\dfrac{79}{3}\right)\sqrt{20}\right)-\left(\left(-6\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(-2\right)\sqrt{25}+\left(6\right)\sqrt{25}+\left(-\dfrac{62}{5}\right)\sqrt{125}\right)-\left(-\dfrac{18}{7}\right)\\
&=&\left(30+\left(\left(-\dfrac{158}{3}\right)\sqrt{5}\right)-\left(\left(-30\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)-10+30+\left(-62\right)\sqrt{5}\right)-\left(-\dfrac{18}{7}\right)\\
&=&\left(50+\left(-\dfrac{343}{6}\right)\sqrt{5}\right)-\left(-\dfrac{18}{7}\right)\\
&=&50+\left(-\dfrac{343}{6}\right)\sqrt{5}+\dfrac{18}{7}\\
&=&\dfrac{368}{7}+\left(-\dfrac{343}{6}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(6\right)\sqrt{25}+\left(\left(-\dfrac{79}{3}\right)\sqrt{20}\right)-\left(\left(-6\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{125}\right)+\left(-2\right)\sqrt{25}+\left(6\right)\sqrt{25}+\left(-\dfrac{62}{5}\right)\sqrt{125}\right)\times\left(-\dfrac{18}{7}\right)\\
&=&\left(30+\left(\left(-\dfrac{158}{3}\right)\sqrt{5}\right)-\left(\left(-30\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{5}\right)-10+30+\left(-62\right)\sqrt{5}\right)\times\left(-\dfrac{18}{7}\right)\\
&=&\left(50+\left(-\dfrac{343}{6}\right)\sqrt{5}\right)\left(-\dfrac{18}{7}\right)\\
&=&-\dfrac{900}{7}+\left(147\right)\sqrt{5}\\
&=&-\dfrac{900}{7}+\left(147\right)\sqrt{5}\\
\end{eqnarray*}