L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{47}{5}\) et \( Y=\left(\left(\dfrac{67}{6}\right)\sqrt{20}+\left(\dfrac{67}{6}\right)\sqrt{20}\right)-\left(\left(-5\right)\sqrt{45}+\left(\dfrac{39}{7}\right)\sqrt{45}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{4}{9}\right)\sqrt{20}+\left(-\dfrac{80}{3}\right)\sqrt{20}+\left(-\dfrac{8}{3}\right)\sqrt{125}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{47}{5}\right)+\left(\left(\left(\dfrac{67}{6}\right)\sqrt{20}+\left(\dfrac{67}{6}\right)\sqrt{20}\right)-\left(\left(-5\right)\sqrt{45}+\left(\dfrac{39}{7}\right)\sqrt{45}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{4}{9}\right)\sqrt{20}+\left(-\dfrac{80}{3}\right)\sqrt{20}+\left(-\dfrac{8}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\dfrac{47}{5}\right)+\left(\left(\left(\dfrac{67}{3}\right)\sqrt{5}+\left(\dfrac{67}{3}\right)\sqrt{5}\right)-\left(\left(-15\right)\sqrt{5}+\left(\dfrac{117}{7}\right)\sqrt{5}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{8}{9}\right)\sqrt{5}+\left(-\dfrac{160}{3}\right)\sqrt{5}+\left(-\dfrac{40}{3}\right)\sqrt{5}\right)\right)\\
&=&\dfrac{47}{5}+\left(\left(\dfrac{67}{3}\right)\sqrt{5}+\left(\dfrac{67}{3}\right)\sqrt{5}\right)-\left(\left(-15\right)\sqrt{5}+\left(\dfrac{117}{7}\right)\sqrt{5}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{8}{9}\right)\sqrt{5}+\left(-\dfrac{160}{3}\right)\sqrt{5}+\left(-\dfrac{40}{3}\right)\sqrt{5}\right)\\
&=&\dfrac{503}{20}+\left(\dfrac{6850}{63}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{47}{5}\right)-\left(\left(\left(\dfrac{67}{6}\right)\sqrt{20}+\left(\dfrac{67}{6}\right)\sqrt{20}\right)-\left(\left(-5\right)\sqrt{45}+\left(\dfrac{39}{7}\right)\sqrt{45}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{4}{9}\right)\sqrt{20}+\left(-\dfrac{80}{3}\right)\sqrt{20}+\left(-\dfrac{8}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\dfrac{47}{5}\right)-\left(\left(\left(\dfrac{67}{3}\right)\sqrt{5}+\left(\dfrac{67}{3}\right)\sqrt{5}\right)-\left(\left(-15\right)\sqrt{5}+\left(\dfrac{117}{7}\right)\sqrt{5}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{8}{9}\right)\sqrt{5}+\left(-\dfrac{160}{3}\right)\sqrt{5}+\left(-\dfrac{40}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{47}{5}\right)-\left(\left(\dfrac{6850}{63}\right)\sqrt{5}+\dfrac{63}{4}\right)\\
&=&\dfrac{47}{5}+\left(-\dfrac{6850}{63}\right)\sqrt{5}-\dfrac{63}{4}\\
&=&-\dfrac{127}{20}+\left(-\dfrac{6850}{63}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{47}{5}\right)\times\left(\left(\left(\dfrac{67}{6}\right)\sqrt{20}+\left(\dfrac{67}{6}\right)\sqrt{20}\right)-\left(\left(-5\right)\sqrt{45}+\left(\dfrac{39}{7}\right)\sqrt{45}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{4}{9}\right)\sqrt{20}+\left(-\dfrac{80}{3}\right)\sqrt{20}+\left(-\dfrac{8}{3}\right)\sqrt{125}\right)\right)\\
&=&\left(\dfrac{47}{5}\right)\times\left(\left(\left(\dfrac{67}{3}\right)\sqrt{5}+\left(\dfrac{67}{3}\right)\sqrt{5}\right)-\left(\left(-15\right)\sqrt{5}+\left(\dfrac{117}{7}\right)\sqrt{5}\right)+7-\left(-\dfrac{35}{4}+\left(\dfrac{8}{9}\right)\sqrt{5}+\left(-\dfrac{160}{3}\right)\sqrt{5}+\left(-\dfrac{40}{3}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{47}{5}\right)\left(\left(\dfrac{6850}{63}\right)\sqrt{5}+\dfrac{63}{4}\right)\\
&=&\left(\dfrac{64390}{63}\right)\sqrt{5}+\dfrac{2961}{20}\\
&=&\left(\dfrac{64390}{63}\right)\sqrt{5}+\dfrac{2961}{20}\\
\end{eqnarray*}