L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{71}{9}\right)\sqrt{50}+\left(3\right)\sqrt{50}+\left(-\dfrac{72}{7}\right)\sqrt{4}\) et \( Y=\left(\left(-\dfrac{63}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{8}\right)-6\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{71}{9}\right)\sqrt{50}+\left(3\right)\sqrt{50}+\left(-\dfrac{72}{7}\right)\sqrt{4}\right)+\left(\left(\left(-\dfrac{63}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{8}\right)-6\right)\\
&=&\left(-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{355}{9}\right)\sqrt{2}+\left(15\right)\sqrt{2}-\dfrac{144}{7}\right)+\left(\left(\left(-\dfrac{63}{2}\right)\sqrt{2}\right)-\left(\left(10\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-6\right)\\
&=&-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{355}{9}\right)\sqrt{2}+\left(15\right)\sqrt{2}-\dfrac{144}{7}+\left(\left(-\dfrac{63}{2}\right)\sqrt{2}\right)-\left(\left(10\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-6\\
&=&-\dfrac{298}{7}+\left(-\dfrac{1187}{18}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{71}{9}\right)\sqrt{50}+\left(3\right)\sqrt{50}+\left(-\dfrac{72}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{63}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{8}\right)-6\right)\\
&=&\left(-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{355}{9}\right)\sqrt{2}+\left(15\right)\sqrt{2}-\dfrac{144}{7}\right)-\left(\left(\left(-\dfrac{63}{2}\right)\sqrt{2}\right)-\left(\left(10\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-6\right)\\
&=&\left(-\dfrac{256}{7}+\left(-\dfrac{220}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{83}{2}\right)\sqrt{2}-6\right)\\
&=&-\dfrac{256}{7}+\left(-\dfrac{220}{9}\right)\sqrt{2}+\left(\dfrac{83}{2}\right)\sqrt{2}+6\\
&=&-\dfrac{214}{7}+\left(\dfrac{307}{18}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{71}{9}\right)\sqrt{50}+\left(3\right)\sqrt{50}+\left(-\dfrac{72}{7}\right)\sqrt{4}\right)\times\left(\left(\left(-\dfrac{63}{4}\right)\sqrt{8}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{8}\right)-6\right)\\
&=&\left(-\dfrac{73}{5}-\dfrac{7}{5}+\left(-\dfrac{355}{9}\right)\sqrt{2}+\left(15\right)\sqrt{2}-\dfrac{144}{7}\right)\times\left(\left(\left(-\dfrac{63}{2}\right)\sqrt{2}\right)-\left(\left(10\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-6\right)\\
&=&\left(-\dfrac{256}{7}+\left(-\dfrac{220}{9}\right)\sqrt{2}\right)\left(\left(-\dfrac{83}{2}\right)\sqrt{2}-6\right)\\
&=&\left(\dfrac{34952}{21}\right)\sqrt{2}+\dfrac{1536}{7}+\left(\dfrac{9130}{9}\right)\sqrt{4}\\
&=&\left(\dfrac{34952}{21}\right)\sqrt{2}+\dfrac{141644}{63}\\
\end{eqnarray*}