L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{58}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{4}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\left(\dfrac{59}{2}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{50}\right)+\left(\dfrac{1}{5}\right)\sqrt{4}\) et \( Y=\left(-\dfrac{27}{8}\right)\sqrt{8}+\left(\left(\dfrac{4}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{18}\right)-\left(\left(1\right)\sqrt{8}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{70}{9}\right)\sqrt{8}\right)\) . 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{58}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{4}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\left(\dfrac{59}{2}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{50}\right)+\left(\dfrac{1}{5}\right)\sqrt{4}\right)+\left(\left(-\dfrac{27}{8}\right)\sqrt{8}+\left(\left(\dfrac{4}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{18}\right)-\left(\left(1\right)\sqrt{8}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{70}{9}\right)\sqrt{8}\right)\right)\\
&=&\left(-\dfrac{116}{3}+39+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(\left(\dfrac{177}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)+\dfrac{2}{5}\right)+\left(\left(-\dfrac{27}{4}\right)\sqrt{2}+\left(\left(\dfrac{8}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{43}{3}\right)\sqrt{2}\right)-\left(\left(2\right)\sqrt{2}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{140}{9}\right)\sqrt{2}\right)\right)\\
&=&-\dfrac{116}{3}+39+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(\left(\dfrac{177}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)+\dfrac{2}{5}+\left(-\dfrac{27}{4}\right)\sqrt{2}+\left(\left(\dfrac{8}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{43}{3}\right)\sqrt{2}\right)-\left(\left(2\right)\sqrt{2}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{140}{9}\right)\sqrt{2}\right)\\
&=&\dfrac{467}{105}+\left(-\dfrac{139}{36}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{4}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\left(\dfrac{59}{2}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{50}\right)+\left(\dfrac{1}{5}\right)\sqrt{4}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{8}+\left(\left(\dfrac{4}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{18}\right)-\left(\left(1\right)\sqrt{8}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{70}{9}\right)\sqrt{8}\right)\right)\\
&=&\left(-\dfrac{116}{3}+39+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(\left(\dfrac{177}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)+\dfrac{2}{5}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{2}+\left(\left(\dfrac{8}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{43}{3}\right)\sqrt{2}\right)-\left(\left(2\right)\sqrt{2}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{140}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{11}{15}+\left(1\right)\sqrt{2}\right)-\left(\left(-\dfrac{175}{36}\right)\sqrt{2}+\dfrac{26}{7}\right)\\
&=&\dfrac{11}{15}+\left(1\right)\sqrt{2}+\left(\dfrac{175}{36}\right)\sqrt{2}-\dfrac{26}{7}\\
&=&-\dfrac{313}{105}+\left(\dfrac{211}{36}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{58}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{4}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\left(\dfrac{59}{2}\right)\sqrt{18}\right)-\left(\left(0\right)\sqrt{50}\right)+\left(\dfrac{1}{5}\right)\sqrt{4}\right)\times\left(\left(-\dfrac{27}{8}\right)\sqrt{8}+\left(\left(\dfrac{4}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{43}{9}\right)\sqrt{18}\right)-\left(\left(1\right)\sqrt{8}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{70}{9}\right)\sqrt{8}\right)\right)\\
&=&\left(-\dfrac{116}{3}+39+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(\left(\dfrac{177}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)+\dfrac{2}{5}\right)\times\left(\left(-\dfrac{27}{4}\right)\sqrt{2}+\left(\left(\dfrac{8}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{43}{3}\right)\sqrt{2}\right)-\left(\left(2\right)\sqrt{2}\right)+\dfrac{26}{7}-\left(\left(-\dfrac{140}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{11}{15}+\left(1\right)\sqrt{2}\right)\left(\left(-\dfrac{175}{36}\right)\sqrt{2}+\dfrac{26}{7}\right)\\
&=&\left(\dfrac{113}{756}\right)\sqrt{2}+\dfrac{286}{105}+\left(-\dfrac{175}{36}\right)\sqrt{4}\\
&=&\left(\dfrac{113}{756}\right)\sqrt{2}-\dfrac{4409}{630}\\
\end{eqnarray*}