L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{53}{4}+\left(-\dfrac{17}{5}\right)\sqrt{45}+3\right)-\left(\left(-\dfrac{61}{5}\right)\sqrt{25}\right)\) et \( Y=\left(\left(-\dfrac{31}{8}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{73}{5}\right)\sqrt{125}\right)-\left(\left(-1\right)\sqrt{20}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{53}{4}+\left(-\dfrac{17}{5}\right)\sqrt{45}+3\right)-\left(\left(-\dfrac{61}{5}\right)\sqrt{25}\right)\right)+\left(\left(\left(-\dfrac{31}{8}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{73}{5}\right)\sqrt{125}\right)-\left(\left(-1\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\dfrac{53}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}+3\right)+61\right)+\left(-\dfrac{155}{8}-\left(\left(\left(-73\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\dfrac{53}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}+3\right)+61-\dfrac{155}{8}-\left(\left(\left(-73\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)\right)\\
&=&\dfrac{463}{8}+\left(\dfrac{304}{5}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{53}{4}+\left(-\dfrac{17}{5}\right)\sqrt{45}+3\right)-\left(\left(-\dfrac{61}{5}\right)\sqrt{25}\right)\right)-\left(\left(\left(-\dfrac{31}{8}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{73}{5}\right)\sqrt{125}\right)-\left(\left(-1\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\dfrac{53}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}+3\right)+61\right)-\left(-\dfrac{155}{8}-\left(\left(\left(-73\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\dfrac{309}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}\right)-\left(-\dfrac{155}{8}+\left(71\right)\sqrt{5}\right)\\
&=&\dfrac{309}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}+\dfrac{155}{8}+\left(-71\right)\sqrt{5}\\
&=&\dfrac{773}{8}+\left(-\dfrac{406}{5}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{53}{4}+\left(-\dfrac{17}{5}\right)\sqrt{45}+3\right)-\left(\left(-\dfrac{61}{5}\right)\sqrt{25}\right)\right)\times\left(\left(\left(-\dfrac{31}{8}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{73}{5}\right)\sqrt{125}\right)-\left(\left(-1\right)\sqrt{20}\right)\right)\right)\\
&=&\left(\left(\dfrac{53}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}+3\right)+61\right)\times\left(-\dfrac{155}{8}-\left(\left(\left(-73\right)\sqrt{5}\right)-\left(\left(-2\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\dfrac{309}{4}+\left(-\dfrac{51}{5}\right)\sqrt{5}\right)\left(-\dfrac{155}{8}+\left(71\right)\sqrt{5}\right)\\
&=&-\dfrac{47895}{32}+\left(\dfrac{45459}{8}\right)\sqrt{5}+\left(-\dfrac{3621}{5}\right)\sqrt{25}\\
&=&-\dfrac{163767}{32}+\left(\dfrac{45459}{8}\right)\sqrt{5}\\
\end{eqnarray*}