L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{23}{5}\right)\sqrt{18}\right)-4-\left(\left(-\dfrac{73}{5}\right)\sqrt{18}\right)+\left(\left(\dfrac{70}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{18}\right)+\dfrac{31}{6}\) et \( Y=\left(-\dfrac{78}{7}\right)\sqrt{4}\) . 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{23}{5}\right)\sqrt{18}\right)-4-\left(\left(-\dfrac{73}{5}\right)\sqrt{18}\right)+\left(\left(\dfrac{70}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{18}\right)+\dfrac{31}{6}\right)+\left(\left(-\dfrac{78}{7}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(\dfrac{69}{5}\right)\sqrt{2}\right)-4-\left(\left(-\dfrac{219}{5}\right)\sqrt{2}\right)+\left(\left(\dfrac{350}{3}\right)\sqrt{2}\right)-\left(\left(-14\right)\sqrt{2}\right)+\dfrac{31}{6}\right)+\left(-\dfrac{156}{7}\right)\\
&=&\left(\left(\dfrac{69}{5}\right)\sqrt{2}\right)-4-\left(\left(-\dfrac{219}{5}\right)\sqrt{2}\right)+\left(\left(\dfrac{350}{3}\right)\sqrt{2}\right)-\left(\left(-14\right)\sqrt{2}\right)+\dfrac{31}{6}-\dfrac{156}{7}\\
&=&\left(\dfrac{2824}{15}\right)\sqrt{2}-\dfrac{887}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{23}{5}\right)\sqrt{18}\right)-4-\left(\left(-\dfrac{73}{5}\right)\sqrt{18}\right)+\left(\left(\dfrac{70}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{18}\right)+\dfrac{31}{6}\right)-\left(\left(-\dfrac{78}{7}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(\dfrac{69}{5}\right)\sqrt{2}\right)-4-\left(\left(-\dfrac{219}{5}\right)\sqrt{2}\right)+\left(\left(\dfrac{350}{3}\right)\sqrt{2}\right)-\left(\left(-14\right)\sqrt{2}\right)+\dfrac{31}{6}\right)-\left(-\dfrac{156}{7}\right)\\
&=&\left(\left(\dfrac{2824}{15}\right)\sqrt{2}+\dfrac{7}{6}\right)-\left(-\dfrac{156}{7}\right)\\
&=&\left(\dfrac{2824}{15}\right)\sqrt{2}+\dfrac{7}{6}+\dfrac{156}{7}\\
&=&\left(\dfrac{2824}{15}\right)\sqrt{2}+\dfrac{985}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{23}{5}\right)\sqrt{18}\right)-4-\left(\left(-\dfrac{73}{5}\right)\sqrt{18}\right)+\left(\left(\dfrac{70}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{18}\right)+\dfrac{31}{6}\right)\times\left(\left(-\dfrac{78}{7}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(\dfrac{69}{5}\right)\sqrt{2}\right)-4-\left(\left(-\dfrac{219}{5}\right)\sqrt{2}\right)+\left(\left(\dfrac{350}{3}\right)\sqrt{2}\right)-\left(\left(-14\right)\sqrt{2}\right)+\dfrac{31}{6}\right)\times\left(-\dfrac{156}{7}\right)\\
&=&\left(\left(\dfrac{2824}{15}\right)\sqrt{2}+\dfrac{7}{6}\right)\left(-\dfrac{156}{7}\right)\\
&=&\left(-\dfrac{146848}{35}\right)\sqrt{2}-26\\
&=&\left(-\dfrac{146848}{35}\right)\sqrt{2}-26\\
\end{eqnarray*}