L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{4}{3}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(-\dfrac{61}{4}\right)\sqrt{50}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{5}{3}\right)\sqrt{8}\right)-1+\left(\left(-7\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{50}\right)-\dfrac{67}{3}\) et \( Y=-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{4}{3}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(-\dfrac{61}{4}\right)\sqrt{50}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{5}{3}\right)\sqrt{8}\right)-1+\left(\left(-7\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{50}\right)-\dfrac{67}{3}\right)+\left(-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(4\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(-\dfrac{305}{4}\right)\sqrt{2}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{10}{3}\right)\sqrt{2}\right)-1+\left(\left(-35\right)\sqrt{2}\right)-\left(\left(-\dfrac{315}{8}\right)\sqrt{2}\right)-\dfrac{67}{3}\right)+\left(-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}-\dfrac{98}{5}\right)\right)\\
&=&\left(4\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(-\dfrac{305}{4}\right)\sqrt{2}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{10}{3}\right)\sqrt{2}\right)-1+\left(\left(-35\right)\sqrt{2}\right)-\left(\left(-\dfrac{315}{8}\right)\sqrt{2}\right)-\dfrac{67}{3}-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}-\dfrac{98}{5}\right)\\
&=&\left(-\dfrac{1213}{24}\right)\sqrt{2}+\dfrac{337}{105}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{4}{3}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(-\dfrac{61}{4}\right)\sqrt{50}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{5}{3}\right)\sqrt{8}\right)-1+\left(\left(-7\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{50}\right)-\dfrac{67}{3}\right)-\left(-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(4\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(-\dfrac{305}{4}\right)\sqrt{2}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{10}{3}\right)\sqrt{2}\right)-1+\left(\left(-35\right)\sqrt{2}\right)-\left(\left(-\dfrac{315}{8}\right)\sqrt{2}\right)-\dfrac{67}{3}\right)-\left(-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}-\dfrac{98}{5}\right)\right)\\
&=&\left(\left(-\dfrac{1213}{24}\right)\sqrt{2}-\dfrac{668}{15}\right)-\left(\dfrac{1671}{35}\right)\\
&=&\left(-\dfrac{1213}{24}\right)\sqrt{2}-\dfrac{668}{15}+-\dfrac{1671}{35}\\
&=&\left(-\dfrac{1213}{24}\right)\sqrt{2}-\dfrac{9689}{105}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{4}{3}\right)\sqrt{18}+\left(7\right)\sqrt{8}+\left(-\dfrac{61}{4}\right)\sqrt{50}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{5}{3}\right)\sqrt{8}\right)-1+\left(\left(-7\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{50}\right)-\dfrac{67}{3}\right)\times\left(-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}+\left(-\dfrac{49}{5}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(4\right)\sqrt{2}+\left(14\right)\sqrt{2}+\left(-\dfrac{305}{4}\right)\sqrt{2}-6-1-\dfrac{71}{5}-\left(\left(-\dfrac{10}{3}\right)\sqrt{2}\right)-1+\left(\left(-35\right)\sqrt{2}\right)-\left(\left(-\dfrac{315}{8}\right)\sqrt{2}\right)-\dfrac{67}{3}\right)\times\left(-\dfrac{20}{7}-\left(-\dfrac{31}{2}-\dfrac{31}{2}-\dfrac{98}{5}\right)\right)\\
&=&\left(\left(-\dfrac{1213}{24}\right)\sqrt{2}-\dfrac{668}{15}\right)\left(\dfrac{1671}{35}\right)\\
&=&\left(-\dfrac{675641}{280}\right)\sqrt{2}-\dfrac{372076}{175}\\
&=&\left(-\dfrac{675641}{280}\right)\sqrt{2}-\dfrac{372076}{175}\\
\end{eqnarray*}