L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(9\right)\sqrt{25}+\left(\left(9\right)\sqrt{20}\right)+\dfrac{52}{7}-1+\left(\dfrac{11}{3}\right)\sqrt{20}+\left(1\right)\sqrt{25}\) et \( Y=\left(\left(-\dfrac{44}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)+\left(-\dfrac{61}{3}\right)\sqrt{20}+\dfrac{54}{5}+\left(\dfrac{29}{3}\right)\sqrt{45}+\dfrac{19}{2}+\left(\dfrac{7}{5}\right)\sqrt{25}+\left(\dfrac{35}{2}\right)\sqrt{45}+\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(9\right)\sqrt{25}+\left(\left(9\right)\sqrt{20}\right)+\dfrac{52}{7}-1+\left(\dfrac{11}{3}\right)\sqrt{20}+\left(1\right)\sqrt{25}\right)+\left(\left(\left(-\dfrac{44}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)+\left(-\dfrac{61}{3}\right)\sqrt{20}+\dfrac{54}{5}+\left(\dfrac{29}{3}\right)\sqrt{45}+\dfrac{19}{2}+\left(\dfrac{7}{5}\right)\sqrt{25}+\left(\dfrac{35}{2}\right)\sqrt{45}+\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(45+\left(\left(18\right)\sqrt{5}\right)+\dfrac{52}{7}-1+\left(\dfrac{22}{3}\right)\sqrt{5}+5\right)+\left(-44-\left(\left(-\dfrac{95}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{105}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)+\left(-\dfrac{122}{3}\right)\sqrt{5}+\dfrac{54}{5}+\left(29\right)\sqrt{5}+\dfrac{19}{2}+7+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)\\
&=&45+\left(\left(18\right)\sqrt{5}\right)+\dfrac{52}{7}-1+\left(\dfrac{22}{3}\right)\sqrt{5}+5-44-\left(\left(-\dfrac{95}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{105}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)+\left(-\dfrac{122}{3}\right)\sqrt{5}+\dfrac{54}{5}+\left(29\right)\sqrt{5}+\dfrac{19}{2}+7+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\\
&=&\dfrac{999}{28}+\left(\dfrac{2059}{21}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(9\right)\sqrt{25}+\left(\left(9\right)\sqrt{20}\right)+\dfrac{52}{7}-1+\left(\dfrac{11}{3}\right)\sqrt{20}+\left(1\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{44}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)+\left(-\dfrac{61}{3}\right)\sqrt{20}+\dfrac{54}{5}+\left(\dfrac{29}{3}\right)\sqrt{45}+\dfrac{19}{2}+\left(\dfrac{7}{5}\right)\sqrt{25}+\left(\dfrac{35}{2}\right)\sqrt{45}+\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(45+\left(\left(18\right)\sqrt{5}\right)+\dfrac{52}{7}-1+\left(\dfrac{22}{3}\right)\sqrt{5}+5\right)-\left(-44-\left(\left(-\dfrac{95}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{105}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)+\left(-\dfrac{122}{3}\right)\sqrt{5}+\dfrac{54}{5}+\left(29\right)\sqrt{5}+\dfrac{19}{2}+7+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{395}{7}+\left(\dfrac{76}{3}\right)\sqrt{5}\right)-\left(-\dfrac{83}{4}+\left(\dfrac{509}{7}\right)\sqrt{5}\right)\\
&=&\dfrac{395}{7}+\left(\dfrac{76}{3}\right)\sqrt{5}+\dfrac{83}{4}+\left(-\dfrac{509}{7}\right)\sqrt{5}\\
&=&\dfrac{2161}{28}+\left(-\dfrac{995}{21}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(9\right)\sqrt{25}+\left(\left(9\right)\sqrt{20}\right)+\dfrac{52}{7}-1+\left(\dfrac{11}{3}\right)\sqrt{20}+\left(1\right)\sqrt{25}\right)\times\left(\left(\left(-\dfrac{44}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{19}{7}\right)\sqrt{125}\right)-\left(\left(-\dfrac{31}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{7}\right)\sqrt{125}\right)+\left(-\dfrac{61}{3}\right)\sqrt{20}+\dfrac{54}{5}+\left(\dfrac{29}{3}\right)\sqrt{45}+\dfrac{19}{2}+\left(\dfrac{7}{5}\right)\sqrt{25}+\left(\dfrac{35}{2}\right)\sqrt{45}+\left(\left(\dfrac{17}{2}\right)\sqrt{45}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{15}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(45+\left(\left(18\right)\sqrt{5}\right)+\dfrac{52}{7}-1+\left(\dfrac{22}{3}\right)\sqrt{5}+5\right)\times\left(-44-\left(\left(-\dfrac{95}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{62}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{105}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{5}\right)+\left(-\dfrac{122}{3}\right)\sqrt{5}+\dfrac{54}{5}+\left(29\right)\sqrt{5}+\dfrac{19}{2}+7+\left(\dfrac{105}{2}\right)\sqrt{5}+\left(\left(\dfrac{51}{2}\right)\sqrt{5}\right)-\dfrac{69}{5}+\dfrac{39}{4}-\left(\left(-\dfrac{45}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{395}{7}+\left(\dfrac{76}{3}\right)\sqrt{5}\right)\left(-\dfrac{83}{4}+\left(\dfrac{509}{7}\right)\sqrt{5}\right)\\
&=&-\dfrac{32785}{28}+\left(\dfrac{525892}{147}\right)\sqrt{5}+\left(\dfrac{38684}{21}\right)\sqrt{25}\\
&=&\dfrac{96475}{12}+\left(\dfrac{525892}{147}\right)\sqrt{5}\\
\end{eqnarray*}