L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{25}{8}\right)\sqrt{18}+\left(\dfrac{68}{9}\right)\sqrt{18}+4+\left(-\dfrac{35}{4}\right)\sqrt{18}+\left(-\dfrac{72}{5}\right)\sqrt{4}+\left(\left(-9\right)\sqrt{18}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{18}\right)+\left(-\dfrac{49}{5}\right)\sqrt{4}\) et \( Y=\left(-\dfrac{43}{9}\right)\sqrt{4}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{25}{8}\right)\sqrt{18}+\left(\dfrac{68}{9}\right)\sqrt{18}+4+\left(-\dfrac{35}{4}\right)\sqrt{18}+\left(-\dfrac{72}{5}\right)\sqrt{4}+\left(\left(-9\right)\sqrt{18}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{18}\right)+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)+\left(\left(-\dfrac{43}{9}\right)\sqrt{4}\right)\\
&=&\left(\left(\dfrac{75}{8}\right)\sqrt{2}+\left(\dfrac{68}{3}\right)\sqrt{2}+4+\left(-\dfrac{105}{4}\right)\sqrt{2}-\dfrac{144}{5}+\left(\left(-27\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{2}\right)-\dfrac{98}{5}\right)+\left(-\dfrac{86}{9}\right)\\
&=&\left(\dfrac{75}{8}\right)\sqrt{2}+\left(\dfrac{68}{3}\right)\sqrt{2}+4+\left(-\dfrac{105}{4}\right)\sqrt{2}-\dfrac{144}{5}+\left(\left(-27\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{2}\right)-\dfrac{98}{5}-\dfrac{86}{9}\\
&=&\left(-\dfrac{383}{24}\right)\sqrt{2}-\dfrac{2428}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{25}{8}\right)\sqrt{18}+\left(\dfrac{68}{9}\right)\sqrt{18}+4+\left(-\dfrac{35}{4}\right)\sqrt{18}+\left(-\dfrac{72}{5}\right)\sqrt{4}+\left(\left(-9\right)\sqrt{18}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{18}\right)+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)-\left(\left(-\dfrac{43}{9}\right)\sqrt{4}\right)\\
&=&\left(\left(\dfrac{75}{8}\right)\sqrt{2}+\left(\dfrac{68}{3}\right)\sqrt{2}+4+\left(-\dfrac{105}{4}\right)\sqrt{2}-\dfrac{144}{5}+\left(\left(-27\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{2}\right)-\dfrac{98}{5}\right)-\left(-\dfrac{86}{9}\right)\\
&=&\left(\left(-\dfrac{383}{24}\right)\sqrt{2}-\dfrac{222}{5}\right)-\left(-\dfrac{86}{9}\right)\\
&=&\left(-\dfrac{383}{24}\right)\sqrt{2}-\dfrac{222}{5}+\dfrac{86}{9}\\
&=&\left(-\dfrac{383}{24}\right)\sqrt{2}-\dfrac{1568}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{25}{8}\right)\sqrt{18}+\left(\dfrac{68}{9}\right)\sqrt{18}+4+\left(-\dfrac{35}{4}\right)\sqrt{18}+\left(-\dfrac{72}{5}\right)\sqrt{4}+\left(\left(-9\right)\sqrt{18}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{18}\right)+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)\times\left(\left(-\dfrac{43}{9}\right)\sqrt{4}\right)\\
&=&\left(\left(\dfrac{75}{8}\right)\sqrt{2}+\left(\dfrac{68}{3}\right)\sqrt{2}+4+\left(-\dfrac{105}{4}\right)\sqrt{2}-\dfrac{144}{5}+\left(\left(-27\right)\sqrt{2}\right)-\left(\left(-\dfrac{21}{4}\right)\sqrt{2}\right)-\dfrac{98}{5}\right)\times\left(-\dfrac{86}{9}\right)\\
&=&\left(\left(-\dfrac{383}{24}\right)\sqrt{2}-\dfrac{222}{5}\right)\left(-\dfrac{86}{9}\right)\\
&=&\left(\dfrac{16469}{108}\right)\sqrt{2}+\dfrac{6364}{15}\\
&=&\left(\dfrac{16469}{108}\right)\sqrt{2}+\dfrac{6364}{15}\\
\end{eqnarray*}