L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(1\right)\sqrt{45}\) et \( Y=\left(\left(1\right)\sqrt{20}+\left(-\dfrac{20}{7}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{125}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(1\right)\sqrt{45}\right)+\left(\left(\left(1\right)\sqrt{20}+\left(-\dfrac{20}{7}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{125}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)+\left(\left(\left(2\right)\sqrt{5}-\dfrac{100}{7}+\left(21\right)\sqrt{5}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{69}{7}\right)\sqrt{5}\right)-\left(\left(19\right)\sqrt{5}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\right)\\
&=&\left(3\right)\sqrt{5}+\left(\left(2\right)\sqrt{5}-\dfrac{100}{7}+\left(21\right)\sqrt{5}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{69}{7}\right)\sqrt{5}\right)-\left(\left(19\right)\sqrt{5}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\\
&=&\left(\dfrac{488}{7}\right)\sqrt{5}-\dfrac{799}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(1\right)\sqrt{45}\right)-\left(\left(\left(1\right)\sqrt{20}+\left(-\dfrac{20}{7}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{125}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)-\left(\left(\left(2\right)\sqrt{5}-\dfrac{100}{7}+\left(21\right)\sqrt{5}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{69}{7}\right)\sqrt{5}\right)-\left(\left(19\right)\sqrt{5}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)-\left(\left(\dfrac{467}{7}\right)\sqrt{5}-\dfrac{799}{28}\right)\\
&=&\left(3\right)\sqrt{5}+\left(-\dfrac{467}{7}\right)\sqrt{5}+\dfrac{799}{28}\\
&=&\left(-\dfrac{446}{7}\right)\sqrt{5}+\dfrac{799}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(1\right)\sqrt{45}\right)\times\left(\left(\left(1\right)\sqrt{20}+\left(-\dfrac{20}{7}\right)\sqrt{25}+\left(\dfrac{21}{5}\right)\sqrt{125}+\left(\dfrac{52}{7}\right)\sqrt{20}\right)-\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{45}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{125}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)\times\left(\left(\left(2\right)\sqrt{5}-\dfrac{100}{7}+\left(21\right)\sqrt{5}+\left(\dfrac{104}{7}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{69}{7}\right)\sqrt{5}\right)-\left(\left(19\right)\sqrt{5}\right)\right)-\left(\dfrac{33}{2}-\dfrac{9}{4}\right)\right)\\
&=&\left(\left(3\right)\sqrt{5}\right)\left(\left(\dfrac{467}{7}\right)\sqrt{5}-\dfrac{799}{28}\right)\\
&=&\left(\dfrac{1401}{7}\right)\sqrt{25}+\left(-\dfrac{2397}{28}\right)\sqrt{5}\\
&=&\dfrac{7005}{7}+\left(-\dfrac{2397}{28}\right)\sqrt{5}\\
\end{eqnarray*}