L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{8}{5}\right)\sqrt{8}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-5\right)\sqrt{8}+\left(4\right)\sqrt{4}\right)\) et \( Y=\left(\dfrac{67}{4}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{8}{5}\right)\sqrt{8}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-5\right)\sqrt{8}+\left(4\right)\sqrt{4}\right)\right)+\left(\left(\dfrac{67}{4}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\dfrac{16}{5}\right)\sqrt{2}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-10\right)\sqrt{2}+8\right)\right)+\left(\left(\dfrac{335}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{16}{5}\right)\sqrt{2}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-10\right)\sqrt{2}+8\right)+\left(\dfrac{335}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{1939}{20}\right)\sqrt{2}-\dfrac{37}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{8}{5}\right)\sqrt{8}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-5\right)\sqrt{8}+\left(4\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{67}{4}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\dfrac{16}{5}\right)\sqrt{2}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-10\right)\sqrt{2}+8\right)\right)-\left(\left(\dfrac{335}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{66}{5}\right)\sqrt{2}-\dfrac{37}{2}\right)-\left(\left(\dfrac{335}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{66}{5}\right)\sqrt{2}-\dfrac{37}{2}+\left(-\dfrac{335}{4}\right)\sqrt{2}\\
&=&\left(-\dfrac{1411}{20}\right)\sqrt{2}-\dfrac{37}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{8}{5}\right)\sqrt{8}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-5\right)\sqrt{8}+\left(4\right)\sqrt{4}\right)\right)\times\left(\left(\dfrac{67}{4}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\dfrac{16}{5}\right)\sqrt{2}\right)-\left(-3-\dfrac{35}{6}+\dfrac{58}{3}+\left(-10\right)\sqrt{2}+8\right)\right)\times\left(\left(\dfrac{335}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{66}{5}\right)\sqrt{2}-\dfrac{37}{2}\right)\left(\left(\dfrac{335}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2211}{2}\right)\sqrt{4}+\left(-\dfrac{12395}{8}\right)\sqrt{2}\\
&=&2211+\left(-\dfrac{12395}{8}\right)\sqrt{2}\\
\end{eqnarray*}