L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(4\right)\sqrt{4}+\left(-\dfrac{52}{9}\right)\sqrt{8}+\left(6\right)\sqrt{50}+\left(-\dfrac{27}{5}\right)\sqrt{50}\) et \( Y=\left(\left(-\dfrac{10}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{4}\right)-\dfrac{15}{8}-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{5}{9}\right)\sqrt{50}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(4\right)\sqrt{4}+\left(-\dfrac{52}{9}\right)\sqrt{8}+\left(6\right)\sqrt{50}+\left(-\dfrac{27}{5}\right)\sqrt{50}\right)+\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{4}\right)-\dfrac{15}{8}-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{5}{9}\right)\sqrt{50}\right)\right)\right)\\
&=&\left(8+\left(-\dfrac{104}{9}\right)\sqrt{2}+\left(30\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)+\left(\left(\left(-\dfrac{20}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{185}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)\right)-\left(-\dfrac{71}{2}+\dfrac{68}{7}-\dfrac{15}{8}+\dfrac{66}{7}-\left(\left(-\dfrac{25}{9}\right)\sqrt{2}\right)\right)\right)\\
&=&8+\left(-\dfrac{104}{9}\right)\sqrt{2}+\left(30\right)\sqrt{2}+\left(-27\right)\sqrt{2}+\left(\left(-\dfrac{20}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{185}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)\right)-\left(-\dfrac{71}{2}+\dfrac{68}{7}-\dfrac{15}{8}+\dfrac{66}{7}-\left(\left(-\dfrac{25}{9}\right)\sqrt{2}\right)\right)\\
&=&\dfrac{1469}{56}+\left(-\dfrac{35201}{840}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(4\right)\sqrt{4}+\left(-\dfrac{52}{9}\right)\sqrt{8}+\left(6\right)\sqrt{50}+\left(-\dfrac{27}{5}\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{4}\right)-\dfrac{15}{8}-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{5}{9}\right)\sqrt{50}\right)\right)\right)\\
&=&\left(8+\left(-\dfrac{104}{9}\right)\sqrt{2}+\left(30\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{20}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{185}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)\right)-\left(-\dfrac{71}{2}+\dfrac{68}{7}-\dfrac{15}{8}+\dfrac{66}{7}-\left(\left(-\dfrac{25}{9}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(8+\left(-\dfrac{77}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{84043}{2520}\right)\sqrt{2}+\dfrac{1021}{56}\right)\\
&=&8+\left(-\dfrac{77}{9}\right)\sqrt{2}+\left(\dfrac{84043}{2520}\right)\sqrt{2}-\dfrac{1021}{56}\\
&=&-\dfrac{573}{56}+\left(\dfrac{62483}{2520}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(4\right)\sqrt{4}+\left(-\dfrac{52}{9}\right)\sqrt{8}+\left(6\right)\sqrt{50}+\left(-\dfrac{27}{5}\right)\sqrt{50}\right)\times\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{37}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)\right)-\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{4}\right)-\dfrac{15}{8}-\left(\left(-\dfrac{33}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{5}{9}\right)\sqrt{50}\right)\right)\right)\\
&=&\left(8+\left(-\dfrac{104}{9}\right)\sqrt{2}+\left(30\right)\sqrt{2}+\left(-27\right)\sqrt{2}\right)\times\left(\left(\left(-\dfrac{20}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{28}{5}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{100}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{185}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)\right)-\left(-\dfrac{71}{2}+\dfrac{68}{7}-\dfrac{15}{8}+\dfrac{66}{7}-\left(\left(-\dfrac{25}{9}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(8+\left(-\dfrac{77}{9}\right)\sqrt{2}\right)\left(\left(-\dfrac{84043}{2520}\right)\sqrt{2}+\dfrac{1021}{56}\right)\\
&=&\left(-\dfrac{118381}{280}\right)\sqrt{2}+\dfrac{1021}{7}+\left(\dfrac{924473}{3240}\right)\sqrt{4}\\
&=&\left(-\dfrac{118381}{280}\right)\sqrt{2}+\dfrac{8125331}{11340}\\
\end{eqnarray*}