L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{49}{9}\right)\sqrt{18}\) et \( Y=\left(\dfrac{34}{5}\right)\sqrt{18}+\left(-\dfrac{72}{7}\right)\sqrt{50}+\dfrac{79}{3}+\left(-\dfrac{13}{2}\right)\sqrt{18}+\left(-\dfrac{3}{2}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{49}{9}\right)\sqrt{18}\right)+\left(\left(\dfrac{34}{5}\right)\sqrt{18}+\left(-\dfrac{72}{7}\right)\sqrt{50}+\dfrac{79}{3}+\left(-\dfrac{13}{2}\right)\sqrt{18}+\left(-\dfrac{3}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{49}{3}\right)\sqrt{2}\right)+\left(\left(\dfrac{102}{5}\right)\sqrt{2}+\left(-\dfrac{360}{7}\right)\sqrt{2}+\dfrac{79}{3}+\left(-\dfrac{39}{2}\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{49}{3}\right)\sqrt{2}+\left(\dfrac{102}{5}\right)\sqrt{2}+\left(-\dfrac{360}{7}\right)\sqrt{2}+\dfrac{79}{3}+\left(-\dfrac{39}{2}\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{4378}{105}\right)\sqrt{2}+\dfrac{79}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{49}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{18}+\left(-\dfrac{72}{7}\right)\sqrt{50}+\dfrac{79}{3}+\left(-\dfrac{13}{2}\right)\sqrt{18}+\left(-\dfrac{3}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{49}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{102}{5}\right)\sqrt{2}+\left(-\dfrac{360}{7}\right)\sqrt{2}+\dfrac{79}{3}+\left(-\dfrac{39}{2}\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{49}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{2031}{35}\right)\sqrt{2}+\dfrac{79}{3}\right)\\
&=&\left(\dfrac{49}{3}\right)\sqrt{2}+\left(\dfrac{2031}{35}\right)\sqrt{2}-\dfrac{79}{3}\\
&=&\left(\dfrac{7808}{105}\right)\sqrt{2}-\dfrac{79}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{49}{9}\right)\sqrt{18}\right)\times\left(\left(\dfrac{34}{5}\right)\sqrt{18}+\left(-\dfrac{72}{7}\right)\sqrt{50}+\dfrac{79}{3}+\left(-\dfrac{13}{2}\right)\sqrt{18}+\left(-\dfrac{3}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{49}{3}\right)\sqrt{2}\right)\times\left(\left(\dfrac{102}{5}\right)\sqrt{2}+\left(-\dfrac{360}{7}\right)\sqrt{2}+\dfrac{79}{3}+\left(-\dfrac{39}{2}\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{49}{3}\right)\sqrt{2}\right)\left(\left(-\dfrac{2031}{35}\right)\sqrt{2}+\dfrac{79}{3}\right)\\
&=&\left(-\dfrac{4739}{5}\right)\sqrt{4}+\left(\dfrac{3871}{9}\right)\sqrt{2}\\
&=&-\dfrac{9478}{5}+\left(\dfrac{3871}{9}\right)\sqrt{2}\\
\end{eqnarray*}