L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-8\right)\sqrt{25}+\dfrac{55}{6}-\dfrac{16}{9}+\left(-9\right)\sqrt{125}+\left(-\dfrac{44}{3}\right)\sqrt{20}+\left(-\dfrac{35}{6}\right)\sqrt{25}\) et \( Y=\left(\left(-\dfrac{52}{5}\right)\sqrt{25}+\left(-\dfrac{3}{4}\right)\sqrt{25}+\left(-3\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{69}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{25}\right)-\dfrac{3}{2}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-8\right)\sqrt{25}+\dfrac{55}{6}-\dfrac{16}{9}+\left(-9\right)\sqrt{125}+\left(-\dfrac{44}{3}\right)\sqrt{20}+\left(-\dfrac{35}{6}\right)\sqrt{25}\right)+\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{25}+\left(-\dfrac{3}{4}\right)\sqrt{25}+\left(-3\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{69}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{25}\right)-\dfrac{3}{2}\right)\right)\\
&=&\left(-40+\dfrac{55}{6}-\dfrac{16}{9}+\left(-45\right)\sqrt{5}+\left(-\dfrac{88}{3}\right)\sqrt{5}-\dfrac{175}{6}\right)+\left(\left(-52-\dfrac{15}{4}+\left(-9\right)\sqrt{5}\right)-\left(\dfrac{345}{4}-\left(\left(\dfrac{128}{3}\right)\sqrt{5}\right)+\dfrac{305}{8}-\dfrac{3}{2}\right)\right)\\
&=&-40+\dfrac{55}{6}-\dfrac{16}{9}+\left(-45\right)\sqrt{5}+\left(-\dfrac{88}{3}\right)\sqrt{5}-\dfrac{175}{6}+\left(-52-\dfrac{15}{4}+\left(-9\right)\sqrt{5}\right)-\left(\dfrac{345}{4}-\left(\left(\dfrac{128}{3}\right)\sqrt{5}\right)+\dfrac{305}{8}-\dfrac{3}{2}\right)\\
&=&-\dfrac{17309}{72}+\left(-\dfrac{122}{3}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-8\right)\sqrt{25}+\dfrac{55}{6}-\dfrac{16}{9}+\left(-9\right)\sqrt{125}+\left(-\dfrac{44}{3}\right)\sqrt{20}+\left(-\dfrac{35}{6}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{25}+\left(-\dfrac{3}{4}\right)\sqrt{25}+\left(-3\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{69}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{25}\right)-\dfrac{3}{2}\right)\right)\\
&=&\left(-40+\dfrac{55}{6}-\dfrac{16}{9}+\left(-45\right)\sqrt{5}+\left(-\dfrac{88}{3}\right)\sqrt{5}-\dfrac{175}{6}\right)-\left(\left(-52-\dfrac{15}{4}+\left(-9\right)\sqrt{5}\right)-\left(\dfrac{345}{4}-\left(\left(\dfrac{128}{3}\right)\sqrt{5}\right)+\dfrac{305}{8}-\dfrac{3}{2}\right)\right)\\
&=&\left(-\dfrac{556}{9}+\left(-\dfrac{223}{3}\right)\sqrt{5}\right)-\left(-\dfrac{1429}{8}+\left(\dfrac{101}{3}\right)\sqrt{5}\right)\\
&=&-\dfrac{556}{9}+\left(-\dfrac{223}{3}\right)\sqrt{5}+\dfrac{1429}{8}+\left(-\dfrac{101}{3}\right)\sqrt{5}\\
&=&\dfrac{8413}{72}+\left(-108\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-8\right)\sqrt{25}+\dfrac{55}{6}-\dfrac{16}{9}+\left(-9\right)\sqrt{125}+\left(-\dfrac{44}{3}\right)\sqrt{20}+\left(-\dfrac{35}{6}\right)\sqrt{25}\right)\times\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{25}+\left(-\dfrac{3}{4}\right)\sqrt{25}+\left(-3\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{69}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{64}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{25}\right)-\dfrac{3}{2}\right)\right)\\
&=&\left(-40+\dfrac{55}{6}-\dfrac{16}{9}+\left(-45\right)\sqrt{5}+\left(-\dfrac{88}{3}\right)\sqrt{5}-\dfrac{175}{6}\right)\times\left(\left(-52-\dfrac{15}{4}+\left(-9\right)\sqrt{5}\right)-\left(\dfrac{345}{4}-\left(\left(\dfrac{128}{3}\right)\sqrt{5}\right)+\dfrac{305}{8}-\dfrac{3}{2}\right)\right)\\
&=&\left(-\dfrac{556}{9}+\left(-\dfrac{223}{3}\right)\sqrt{5}\right)\left(-\dfrac{1429}{8}+\left(\dfrac{101}{3}\right)\sqrt{5}\right)\\
&=&\dfrac{198631}{18}+\left(\dfrac{2418755}{216}\right)\sqrt{5}+\left(-\dfrac{22523}{9}\right)\sqrt{25}\\
&=&-\dfrac{26599}{18}+\left(\dfrac{2418755}{216}\right)\sqrt{5}\\
\end{eqnarray*}