L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{72}{5}\right)\sqrt{50}\) et \( Y=\left(-\dfrac{44}{5}\right)\sqrt{18}+\dfrac{79}{4}-\left(\left(-\dfrac{20}{3}\right)\sqrt{18}\right)+\dfrac{16}{9}-\left(\left(\dfrac{53}{9}\right)\sqrt{50}\right)+\left(\dfrac{11}{3}\right)\sqrt{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{72}{5}\right)\sqrt{50}\right)+\left(\left(-\dfrac{44}{5}\right)\sqrt{18}+\dfrac{79}{4}-\left(\left(-\dfrac{20}{3}\right)\sqrt{18}\right)+\dfrac{16}{9}-\left(\left(\dfrac{53}{9}\right)\sqrt{50}\right)+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(72\right)\sqrt{2}\right)+\left(\left(-\dfrac{132}{5}\right)\sqrt{2}+\dfrac{79}{4}-\left(\left(-20\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(\dfrac{265}{9}\right)\sqrt{2}\right)+\left(\dfrac{22}{3}\right)\sqrt{2}\right)\\
&=&\left(72\right)\sqrt{2}+\left(-\dfrac{132}{5}\right)\sqrt{2}+\dfrac{79}{4}-\left(\left(-20\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(\dfrac{265}{9}\right)\sqrt{2}\right)+\left(\dfrac{22}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{1957}{45}\right)\sqrt{2}+\dfrac{775}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{72}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{44}{5}\right)\sqrt{18}+\dfrac{79}{4}-\left(\left(-\dfrac{20}{3}\right)\sqrt{18}\right)+\dfrac{16}{9}-\left(\left(\dfrac{53}{9}\right)\sqrt{50}\right)+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(72\right)\sqrt{2}\right)-\left(\left(-\dfrac{132}{5}\right)\sqrt{2}+\dfrac{79}{4}-\left(\left(-20\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(\dfrac{265}{9}\right)\sqrt{2}\right)+\left(\dfrac{22}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(72\right)\sqrt{2}\right)-\left(\left(-\dfrac{1283}{45}\right)\sqrt{2}+\dfrac{775}{36}\right)\\
&=&\left(72\right)\sqrt{2}+\left(\dfrac{1283}{45}\right)\sqrt{2}-\dfrac{775}{36}\\
&=&\left(\dfrac{4523}{45}\right)\sqrt{2}-\dfrac{775}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{72}{5}\right)\sqrt{50}\right)\times\left(\left(-\dfrac{44}{5}\right)\sqrt{18}+\dfrac{79}{4}-\left(\left(-\dfrac{20}{3}\right)\sqrt{18}\right)+\dfrac{16}{9}-\left(\left(\dfrac{53}{9}\right)\sqrt{50}\right)+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(72\right)\sqrt{2}\right)\times\left(\left(-\dfrac{132}{5}\right)\sqrt{2}+\dfrac{79}{4}-\left(\left(-20\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(\dfrac{265}{9}\right)\sqrt{2}\right)+\left(\dfrac{22}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(72\right)\sqrt{2}\right)\left(\left(-\dfrac{1283}{45}\right)\sqrt{2}+\dfrac{775}{36}\right)\\
&=&\left(-\dfrac{10264}{5}\right)\sqrt{4}+\left(1550\right)\sqrt{2}\\
&=&-\dfrac{20528}{5}+\left(1550\right)\sqrt{2}\\
\end{eqnarray*}