L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-8\right)\sqrt{18}+\dfrac{17}{3}-\left(\left(\dfrac{74}{3}\right)\sqrt{50}\right)-\left(\left(-5\right)\sqrt{18}\right)+\left(\left(\dfrac{69}{7}\right)\sqrt{4}\right)-\left(\left(-4\right)\sqrt{50}\right)\) et \( Y=-\dfrac{11}{7}+\left(-2\right)\sqrt{4}+\left(-\dfrac{37}{2}\right)\sqrt{8}-\dfrac{64}{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{8}\right)+\dfrac{49}{9}\) . 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{18}+\dfrac{17}{3}-\left(\left(\dfrac{74}{3}\right)\sqrt{50}\right)-\left(\left(-5\right)\sqrt{18}\right)+\left(\left(\dfrac{69}{7}\right)\sqrt{4}\right)-\left(\left(-4\right)\sqrt{50}\right)\right)+\left(-\dfrac{11}{7}+\left(-2\right)\sqrt{4}+\left(-\dfrac{37}{2}\right)\sqrt{8}-\dfrac{64}{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{8}\right)+\dfrac{49}{9}\right)\\
&=&\left(\left(-24\right)\sqrt{2}+\dfrac{17}{3}-\left(\left(\dfrac{370}{3}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)+\dfrac{138}{7}-\left(\left(-20\right)\sqrt{2}\right)\right)+\left(-\dfrac{11}{7}-4+\left(-37\right)\sqrt{2}-\dfrac{64}{5}+\left(\left(-\dfrac{58}{3}\right)\sqrt{2}\right)+\dfrac{49}{9}\right)\\
&=&\left(-24\right)\sqrt{2}+\dfrac{17}{3}-\left(\left(\dfrac{370}{3}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)+\dfrac{138}{7}-\left(\left(-20\right)\sqrt{2}\right)-\dfrac{11}{7}-4+\left(-37\right)\sqrt{2}-\dfrac{64}{5}+\left(\left(-\dfrac{58}{3}\right)\sqrt{2}\right)+\dfrac{49}{9}\\
&=&\left(-\dfrac{506}{3}\right)\sqrt{2}+\dfrac{3923}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-8\right)\sqrt{18}+\dfrac{17}{3}-\left(\left(\dfrac{74}{3}\right)\sqrt{50}\right)-\left(\left(-5\right)\sqrt{18}\right)+\left(\left(\dfrac{69}{7}\right)\sqrt{4}\right)-\left(\left(-4\right)\sqrt{50}\right)\right)-\left(-\dfrac{11}{7}+\left(-2\right)\sqrt{4}+\left(-\dfrac{37}{2}\right)\sqrt{8}-\dfrac{64}{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{8}\right)+\dfrac{49}{9}\right)\\
&=&\left(\left(-24\right)\sqrt{2}+\dfrac{17}{3}-\left(\left(\dfrac{370}{3}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)+\dfrac{138}{7}-\left(\left(-20\right)\sqrt{2}\right)\right)-\left(-\dfrac{11}{7}-4+\left(-37\right)\sqrt{2}-\dfrac{64}{5}+\left(\left(-\dfrac{58}{3}\right)\sqrt{2}\right)+\dfrac{49}{9}\right)\\
&=&\left(\left(-\dfrac{337}{3}\right)\sqrt{2}+\dfrac{533}{21}\right)-\left(-\dfrac{4072}{315}+\left(-\dfrac{169}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{337}{3}\right)\sqrt{2}+\dfrac{533}{21}+\dfrac{4072}{315}+\left(\dfrac{169}{3}\right)\sqrt{2}\\
&=&\left(-56\right)\sqrt{2}+\dfrac{12067}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-8\right)\sqrt{18}+\dfrac{17}{3}-\left(\left(\dfrac{74}{3}\right)\sqrt{50}\right)-\left(\left(-5\right)\sqrt{18}\right)+\left(\left(\dfrac{69}{7}\right)\sqrt{4}\right)-\left(\left(-4\right)\sqrt{50}\right)\right)\times\left(-\dfrac{11}{7}+\left(-2\right)\sqrt{4}+\left(-\dfrac{37}{2}\right)\sqrt{8}-\dfrac{64}{5}+\left(\left(-\dfrac{29}{3}\right)\sqrt{8}\right)+\dfrac{49}{9}\right)\\
&=&\left(\left(-24\right)\sqrt{2}+\dfrac{17}{3}-\left(\left(\dfrac{370}{3}\right)\sqrt{2}\right)-\left(\left(-15\right)\sqrt{2}\right)+\dfrac{138}{7}-\left(\left(-20\right)\sqrt{2}\right)\right)\times\left(-\dfrac{11}{7}-4+\left(-37\right)\sqrt{2}-\dfrac{64}{5}+\left(\left(-\dfrac{58}{3}\right)\sqrt{2}\right)+\dfrac{49}{9}\right)\\
&=&\left(\left(-\dfrac{337}{3}\right)\sqrt{2}+\dfrac{533}{21}\right)\left(-\dfrac{4072}{315}+\left(-\dfrac{169}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{21109}{945}\right)\sqrt{2}+\left(\dfrac{56953}{9}\right)\sqrt{4}-\dfrac{2170376}{6615}\\
&=&\left(\dfrac{21109}{945}\right)\sqrt{2}+\dfrac{81550534}{6615}\\
\end{eqnarray*}