L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{33}{8}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{18}+\left(\dfrac{22}{7}\right)\sqrt{8}+\dfrac{80}{9}+\left(-\dfrac{65}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{18}-1\right)\) et \( Y=\left(-\dfrac{41}{8}\right)\sqrt{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{33}{8}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{18}+\left(\dfrac{22}{7}\right)\sqrt{8}+\dfrac{80}{9}+\left(-\dfrac{65}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{18}-1\right)\right)+\left(\left(-\dfrac{41}{8}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{99}{8}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)-\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{44}{7}\right)\sqrt{2}+\dfrac{80}{9}+\left(-\dfrac{325}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}+\left(\dfrac{129}{5}\right)\sqrt{2}-1\right)\right)+\left(\left(-\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{99}{8}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)-\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{44}{7}\right)\sqrt{2}+\dfrac{80}{9}+\left(-\dfrac{325}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}+\left(\dfrac{129}{5}\right)\sqrt{2}-1\right)+\left(-\dfrac{41}{4}\right)\sqrt{2}\\
&=&\left(-\dfrac{37001}{2520}\right)\sqrt{2}-\dfrac{71}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{33}{8}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{18}+\left(\dfrac{22}{7}\right)\sqrt{8}+\dfrac{80}{9}+\left(-\dfrac{65}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{18}-1\right)\right)-\left(\left(-\dfrac{41}{8}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{99}{8}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)-\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{44}{7}\right)\sqrt{2}+\dfrac{80}{9}+\left(-\dfrac{325}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}+\left(\dfrac{129}{5}\right)\sqrt{2}-1\right)\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{11171}{2520}\right)\sqrt{2}-\dfrac{71}{9}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{11171}{2520}\right)\sqrt{2}-\dfrac{71}{9}+\left(\dfrac{41}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{14659}{2520}\right)\sqrt{2}-\dfrac{71}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{33}{8}\right)\sqrt{18}+\left(0\right)\sqrt{18}\right)-\left(\left(8\right)\sqrt{18}\right)-\left(\left(\dfrac{8}{3}\right)\sqrt{8}\right)-\left(\left(-2\right)\sqrt{18}+\left(\dfrac{22}{7}\right)\sqrt{8}+\dfrac{80}{9}+\left(-\dfrac{65}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{8}+\left(\dfrac{43}{5}\right)\sqrt{18}-1\right)\right)\times\left(\left(-\dfrac{41}{8}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{99}{8}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)-\left(\left(24\right)\sqrt{2}\right)-\left(\left(\dfrac{16}{3}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}+\left(\dfrac{44}{7}\right)\sqrt{2}+\dfrac{80}{9}+\left(-\dfrac{325}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}+\left(\dfrac{129}{5}\right)\sqrt{2}-1\right)\right)\times\left(\left(-\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{11171}{2520}\right)\sqrt{2}-\dfrac{71}{9}\right)\left(\left(-\dfrac{41}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{458011}{10080}\right)\sqrt{4}+\left(\dfrac{2911}{36}\right)\sqrt{2}\\
&=&\dfrac{458011}{5040}+\left(\dfrac{2911}{36}\right)\sqrt{2}\\
\end{eqnarray*}