L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\dfrac{43}{7}\) et \( Y=\left(\left(-\dfrac{40}{9}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{51}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{4}\right)+\dfrac{46}{3}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(-\dfrac{33}{2}\right)\sqrt{50}+\left(-\dfrac{76}{7}\right)\sqrt{4}+\left(-\dfrac{77}{8}\right)\sqrt{18}+\left(\dfrac{5}{2}\right)\sqrt{18}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{43}{7}\right)+\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{51}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{4}\right)+\dfrac{46}{3}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(-\dfrac{33}{2}\right)\sqrt{50}+\left(-\dfrac{76}{7}\right)\sqrt{4}+\left(-\dfrac{77}{8}\right)\sqrt{18}+\left(\dfrac{5}{2}\right)\sqrt{18}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{43}{7}\right)+\left(-\dfrac{80}{9}-\left(51-\left(\left(11\right)\sqrt{2}\right)-7+\dfrac{46}{3}\right)-\left(\dfrac{144}{7}+\left(-\dfrac{165}{2}\right)\sqrt{2}-\dfrac{152}{7}+\left(-\dfrac{231}{8}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)\right)\right)\\
&=&\dfrac{43}{7}-\dfrac{80}{9}-\left(51-\left(\left(11\right)\sqrt{2}\right)-7+\dfrac{46}{3}\right)-\left(\dfrac{144}{7}+\left(-\dfrac{165}{2}\right)\sqrt{2}-\dfrac{152}{7}+\left(-\dfrac{231}{8}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)\right)\\
&=&-\dfrac{19069}{315}+\left(\dfrac{811}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{43}{7}\right)-\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{51}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{4}\right)+\dfrac{46}{3}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(-\dfrac{33}{2}\right)\sqrt{50}+\left(-\dfrac{76}{7}\right)\sqrt{4}+\left(-\dfrac{77}{8}\right)\sqrt{18}+\left(\dfrac{5}{2}\right)\sqrt{18}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{43}{7}\right)-\left(-\dfrac{80}{9}-\left(51-\left(\left(11\right)\sqrt{2}\right)-7+\dfrac{46}{3}\right)-\left(\dfrac{144}{7}+\left(-\dfrac{165}{2}\right)\sqrt{2}-\dfrac{152}{7}+\left(-\dfrac{231}{8}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{43}{7}\right)-\left(-\dfrac{21004}{315}+\left(\dfrac{811}{8}\right)\sqrt{2}\right)\\
&=&\dfrac{43}{7}+\dfrac{21004}{315}+\left(-\dfrac{811}{8}\right)\sqrt{2}\\
&=&\dfrac{3277}{45}+\left(-\dfrac{811}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{43}{7}\right)\times\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{51}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{4}\right)+\dfrac{46}{3}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(-\dfrac{33}{2}\right)\sqrt{50}+\left(-\dfrac{76}{7}\right)\sqrt{4}+\left(-\dfrac{77}{8}\right)\sqrt{18}+\left(\dfrac{5}{2}\right)\sqrt{18}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\dfrac{43}{7}\right)\times\left(-\dfrac{80}{9}-\left(51-\left(\left(11\right)\sqrt{2}\right)-7+\dfrac{46}{3}\right)-\left(\dfrac{144}{7}+\left(-\dfrac{165}{2}\right)\sqrt{2}-\dfrac{152}{7}+\left(-\dfrac{231}{8}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(-\dfrac{2}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{43}{7}\right)\left(-\dfrac{21004}{315}+\left(\dfrac{811}{8}\right)\sqrt{2}\right)\\
&=&-\dfrac{903172}{2205}+\left(\dfrac{34873}{56}\right)\sqrt{2}\\
&=&-\dfrac{903172}{2205}+\left(\dfrac{34873}{56}\right)\sqrt{2}\\
\end{eqnarray*}