L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{47}{6}\right)\sqrt{125}\right)+\dfrac{71}{9}-\left(\left(-\dfrac{35}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{25}\right)+\dfrac{63}{8}+\left(-6\right)\sqrt{45}+\left(-8\right)\sqrt{25}\) et \( Y=\left(\left(\dfrac{27}{2}\right)\sqrt{25}+\dfrac{77}{4}+\left(3\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{73}{6}\right)\sqrt{125}+4+\left(\dfrac{10}{9}\right)\sqrt{25}+\left(-\dfrac{61}{9}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{56}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)+1+\dfrac{38}{3}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{47}{6}\right)\sqrt{125}\right)+\dfrac{71}{9}-\left(\left(-\dfrac{35}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{25}\right)+\dfrac{63}{8}+\left(-6\right)\sqrt{45}+\left(-8\right)\sqrt{25}\right)+\left(\left(\left(\dfrac{27}{2}\right)\sqrt{25}+\dfrac{77}{4}+\left(3\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{73}{6}\right)\sqrt{125}+4+\left(\dfrac{10}{9}\right)\sqrt{25}+\left(-\dfrac{61}{9}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{56}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)+1+\dfrac{38}{3}\right)\\
&=&\left(\left(\left(-\dfrac{235}{6}\right)\sqrt{5}\right)+\dfrac{71}{9}+\dfrac{175}{4}+10+\dfrac{63}{8}+\left(-18\right)\sqrt{5}-40\right)+\left(\left(\dfrac{135}{2}+\dfrac{77}{4}+\left(6\right)\sqrt{5}\right)-\left(-35+\left(-\dfrac{365}{6}\right)\sqrt{5}+4+\dfrac{50}{9}+\left(-\dfrac{61}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{315}{8}\right)+1+\dfrac{38}{3}\right)\\
&=&\left(\left(-\dfrac{235}{6}\right)\sqrt{5}\right)+\dfrac{71}{9}+\dfrac{175}{4}+10+\dfrac{63}{8}+\left(-18\right)\sqrt{5}-40+\left(\dfrac{135}{2}+\dfrac{77}{4}+\left(6\right)\sqrt{5}\right)-\left(-35+\left(-\dfrac{365}{6}\right)\sqrt{5}+4+\dfrac{50}{9}+\left(-\dfrac{61}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{315}{8}\right)+1+\dfrac{38}{3}\\
&=&\left(\dfrac{370}{3}\right)\sqrt{5}+\dfrac{779}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{47}{6}\right)\sqrt{125}\right)+\dfrac{71}{9}-\left(\left(-\dfrac{35}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{25}\right)+\dfrac{63}{8}+\left(-6\right)\sqrt{45}+\left(-8\right)\sqrt{25}\right)-\left(\left(\left(\dfrac{27}{2}\right)\sqrt{25}+\dfrac{77}{4}+\left(3\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{73}{6}\right)\sqrt{125}+4+\left(\dfrac{10}{9}\right)\sqrt{25}+\left(-\dfrac{61}{9}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{56}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)+1+\dfrac{38}{3}\right)\\
&=&\left(\left(\left(-\dfrac{235}{6}\right)\sqrt{5}\right)+\dfrac{71}{9}+\dfrac{175}{4}+10+\dfrac{63}{8}+\left(-18\right)\sqrt{5}-40\right)-\left(\left(\dfrac{135}{2}+\dfrac{77}{4}+\left(6\right)\sqrt{5}\right)-\left(-35+\left(-\dfrac{365}{6}\right)\sqrt{5}+4+\dfrac{50}{9}+\left(-\dfrac{61}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{315}{8}\right)+1+\dfrac{38}{3}\right)\\
&=&\left(\left(-\dfrac{343}{6}\right)\sqrt{5}+\dfrac{2125}{72}\right)-\left(\dfrac{11897}{72}+\left(\dfrac{361}{2}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{343}{6}\right)\sqrt{5}+\dfrac{2125}{72}+-\dfrac{11897}{72}+\left(-\dfrac{361}{2}\right)\sqrt{5}\\
&=&\left(-\dfrac{713}{3}\right)\sqrt{5}-\dfrac{2443}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{47}{6}\right)\sqrt{125}\right)+\dfrac{71}{9}-\left(\left(-\dfrac{35}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{25}\right)+\dfrac{63}{8}+\left(-6\right)\sqrt{45}+\left(-8\right)\sqrt{25}\right)\times\left(\left(\left(\dfrac{27}{2}\right)\sqrt{25}+\dfrac{77}{4}+\left(3\right)\sqrt{20}\right)-\left(\left(-7\right)\sqrt{25}+\left(-\dfrac{73}{6}\right)\sqrt{125}+4+\left(\dfrac{10}{9}\right)\sqrt{25}+\left(-\dfrac{61}{9}\right)\sqrt{45}\right)-\left(\left(\left(-\dfrac{56}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{63}{8}\right)\sqrt{25}\right)\right)+1+\dfrac{38}{3}\right)\\
&=&\left(\left(\left(-\dfrac{235}{6}\right)\sqrt{5}\right)+\dfrac{71}{9}+\dfrac{175}{4}+10+\dfrac{63}{8}+\left(-18\right)\sqrt{5}-40\right)\times\left(\left(\dfrac{135}{2}+\dfrac{77}{4}+\left(6\right)\sqrt{5}\right)-\left(-35+\left(-\dfrac{365}{6}\right)\sqrt{5}+4+\dfrac{50}{9}+\left(-\dfrac{61}{3}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{280}{3}\right)\sqrt{5}\right)-\dfrac{315}{8}\right)+1+\dfrac{38}{3}\right)\\
&=&\left(\left(-\dfrac{343}{6}\right)\sqrt{5}+\dfrac{2125}{72}\right)\left(\dfrac{11897}{72}+\left(\dfrac{361}{2}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{111206}{27}\right)\sqrt{5}+\left(-\dfrac{123823}{12}\right)\sqrt{25}+\dfrac{25281125}{5184}\\
&=&\left(-\dfrac{111206}{27}\right)\sqrt{5}-\dfrac{2906118660}{62208}\\
\end{eqnarray*}