L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{15}{8}\right)\sqrt{4}+\left(-\dfrac{73}{5}\right)\sqrt{18}+\left(0\right)\sqrt{50}+\left(-\dfrac{79}{5}\right)\sqrt{4}-\dfrac{7}{2}\) et \( Y=\left(\left(\left(\dfrac{63}{4}\right)\sqrt{18}\right)+\dfrac{67}{2}-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(0\right)\sqrt{50}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{50}+\left(\dfrac{1}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{4}\right)\) . 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{15}{8}\right)\sqrt{4}+\left(-\dfrac{73}{5}\right)\sqrt{18}+\left(0\right)\sqrt{50}+\left(-\dfrac{79}{5}\right)\sqrt{4}-\dfrac{7}{2}\right)+\left(\left(\left(\left(\dfrac{63}{4}\right)\sqrt{18}\right)+\dfrac{67}{2}-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(0\right)\sqrt{50}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{50}+\left(\dfrac{1}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{4}\right)\right)\\
&=&\left(\dfrac{15}{4}+\left(-\dfrac{219}{5}\right)\sqrt{2}+\left(0\right)\sqrt{2}-\dfrac{158}{5}-\dfrac{7}{2}\right)+\left(\left(\left(\left(\dfrac{189}{4}\right)\sqrt{2}\right)+\dfrac{67}{2}-0\right)-\left(\left(0\right)\sqrt{2}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{2}+\left(\dfrac{5}{2}\right)\sqrt{2}-2\right)\right)\\
&=&\dfrac{15}{4}+\left(-\dfrac{219}{5}\right)\sqrt{2}+\left(0\right)\sqrt{2}-\dfrac{158}{5}-\dfrac{7}{2}+\left(\left(\left(\dfrac{189}{4}\right)\sqrt{2}\right)+\dfrac{67}{2}-0\right)-\left(\left(0\right)\sqrt{2}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{2}+\left(\dfrac{5}{2}\right)\sqrt{2}-2\right)\\
&=&-\dfrac{49}{40}+\left(\dfrac{19}{20}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{15}{8}\right)\sqrt{4}+\left(-\dfrac{73}{5}\right)\sqrt{18}+\left(0\right)\sqrt{50}+\left(-\dfrac{79}{5}\right)\sqrt{4}-\dfrac{7}{2}\right)-\left(\left(\left(\left(\dfrac{63}{4}\right)\sqrt{18}\right)+\dfrac{67}{2}-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(0\right)\sqrt{50}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{50}+\left(\dfrac{1}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{4}\right)\right)\\
&=&\left(\dfrac{15}{4}+\left(-\dfrac{219}{5}\right)\sqrt{2}+\left(0\right)\sqrt{2}-\dfrac{158}{5}-\dfrac{7}{2}\right)-\left(\left(\left(\left(\dfrac{189}{4}\right)\sqrt{2}\right)+\dfrac{67}{2}-0\right)-\left(\left(0\right)\sqrt{2}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{2}+\left(\dfrac{5}{2}\right)\sqrt{2}-2\right)\right)\\
&=&\left(-\dfrac{627}{20}+\left(-\dfrac{219}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{179}{4}\right)\sqrt{2}+\dfrac{241}{8}\right)\\
&=&-\dfrac{627}{20}+\left(-\dfrac{219}{5}\right)\sqrt{2}+\left(-\dfrac{179}{4}\right)\sqrt{2}-\dfrac{241}{8}\\
&=&-\dfrac{2459}{40}+\left(-\dfrac{1771}{20}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{15}{8}\right)\sqrt{4}+\left(-\dfrac{73}{5}\right)\sqrt{18}+\left(0\right)\sqrt{50}+\left(-\dfrac{79}{5}\right)\sqrt{4}-\dfrac{7}{2}\right)\times\left(\left(\left(\left(\dfrac{63}{4}\right)\sqrt{18}\right)+\dfrac{67}{2}-\left(\left(0\right)\sqrt{4}\right)\right)-\left(\left(0\right)\sqrt{50}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{50}+\left(\dfrac{1}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{4}\right)\right)\\
&=&\left(\dfrac{15}{4}+\left(-\dfrac{219}{5}\right)\sqrt{2}+\left(0\right)\sqrt{2}-\dfrac{158}{5}-\dfrac{7}{2}\right)\times\left(\left(\left(\left(\dfrac{189}{4}\right)\sqrt{2}\right)+\dfrac{67}{2}-0\right)-\left(\left(0\right)\sqrt{2}+\dfrac{43}{8}\right)-\left(\left(0\right)\sqrt{2}+\left(\dfrac{5}{2}\right)\sqrt{2}-2\right)\right)\\
&=&\left(-\dfrac{627}{20}+\left(-\dfrac{219}{5}\right)\sqrt{2}\right)\left(\left(\dfrac{179}{4}\right)\sqrt{2}+\dfrac{241}{8}\right)\\
&=&\left(-\dfrac{217791}{80}\right)\sqrt{2}-\dfrac{151107}{160}+\left(-\dfrac{39201}{20}\right)\sqrt{4}\\
&=&\left(-\dfrac{217791}{80}\right)\sqrt{2}-\dfrac{778323}{160}\\
\end{eqnarray*}