L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{56}{5}\right)\sqrt{175}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{4}\right)\sqrt{28}\right)-\dfrac{46}{7}\) et \( Y=\left(\left(6\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{27}{4}\right)\sqrt{49}\right)-\dfrac{43}{8}-\left(\left(-6\right)\sqrt{49}\right)-\left(\left(\dfrac{13}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{175}\right)\right)-\left(\left(4\right)\sqrt{63}\right)-\left(-5-\left(\left(\dfrac{17}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)\right)-\dfrac{61}{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{56}{5}\right)\sqrt{175}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{4}\right)\sqrt{28}\right)-\dfrac{46}{7}\right)+\left(\left(\left(6\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{27}{4}\right)\sqrt{49}\right)-\dfrac{43}{8}-\left(\left(-6\right)\sqrt{49}\right)-\left(\left(\dfrac{13}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{175}\right)\right)-\left(\left(4\right)\sqrt{63}\right)-\left(-5-\left(\left(\dfrac{17}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)\right)-\dfrac{61}{9}\right)\\
&=&\left(\left(\left(-56\right)\sqrt{7}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{2}\right)\sqrt{7}\right)-\dfrac{46}{7}\right)+\left(\left(\left(18\right)\sqrt{7}\right)-\left(\dfrac{189}{4}-\dfrac{43}{8}+42-\dfrac{91}{6}-\left(\left(\dfrac{100}{9}\right)\sqrt{7}\right)\right)-\left(\left(12\right)\sqrt{7}\right)-\left(-5-\left(\left(\dfrac{17}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{280}{3}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{61}{9}\right)\\
&=&\left(\left(-56\right)\sqrt{7}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{2}\right)\sqrt{7}\right)-\dfrac{46}{7}+\left(\left(18\right)\sqrt{7}\right)-\left(\dfrac{189}{4}-\dfrac{43}{8}+42-\dfrac{91}{6}-\left(\left(\dfrac{100}{9}\right)\sqrt{7}\right)\right)-\left(\left(12\right)\sqrt{7}\right)-\left(-5-\left(\left(\dfrac{17}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{280}{3}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{61}{9}\\
&=&\left(\dfrac{1523}{18}\right)\sqrt{7}-\dfrac{35029}{504}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{56}{5}\right)\sqrt{175}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{4}\right)\sqrt{28}\right)-\dfrac{46}{7}\right)-\left(\left(\left(6\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{27}{4}\right)\sqrt{49}\right)-\dfrac{43}{8}-\left(\left(-6\right)\sqrt{49}\right)-\left(\left(\dfrac{13}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{175}\right)\right)-\left(\left(4\right)\sqrt{63}\right)-\left(-5-\left(\left(\dfrac{17}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)\right)-\dfrac{61}{9}\right)\\
&=&\left(\left(\left(-56\right)\sqrt{7}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{2}\right)\sqrt{7}\right)-\dfrac{46}{7}\right)-\left(\left(\left(18\right)\sqrt{7}\right)-\left(\dfrac{189}{4}-\dfrac{43}{8}+42-\dfrac{91}{6}-\left(\left(\dfrac{100}{9}\right)\sqrt{7}\right)\right)-\left(\left(12\right)\sqrt{7}\right)-\left(-5-\left(\left(\dfrac{17}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{280}{3}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{61}{9}\right)\\
&=&\left(\left(-\dfrac{73}{2}\right)\sqrt{7}+\dfrac{62}{63}\right)-\left(\left(\dfrac{1090}{9}\right)\sqrt{7}-\dfrac{5075}{72}\right)\\
&=&\left(-\dfrac{73}{2}\right)\sqrt{7}+\dfrac{62}{63}+\left(-\dfrac{1090}{9}\right)\sqrt{7}+\dfrac{5075}{72}\\
&=&\left(-\dfrac{2837}{18}\right)\sqrt{7}+\dfrac{12007}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{56}{5}\right)\sqrt{175}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{4}\right)\sqrt{28}\right)-\dfrac{46}{7}\right)\times\left(\left(\left(6\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{27}{4}\right)\sqrt{49}\right)-\dfrac{43}{8}-\left(\left(-6\right)\sqrt{49}\right)-\left(\left(\dfrac{13}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{20}{9}\right)\sqrt{175}\right)\right)-\left(\left(4\right)\sqrt{63}\right)-\left(-5-\left(\left(\dfrac{17}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)\right)-\dfrac{61}{9}\right)\\
&=&\left(\left(\left(-56\right)\sqrt{7}\right)+\dfrac{68}{9}+\left(\left(\dfrac{39}{2}\right)\sqrt{7}\right)-\dfrac{46}{7}\right)\times\left(\left(\left(18\right)\sqrt{7}\right)-\left(\dfrac{189}{4}-\dfrac{43}{8}+42-\dfrac{91}{6}-\left(\left(\dfrac{100}{9}\right)\sqrt{7}\right)\right)-\left(\left(12\right)\sqrt{7}\right)-\left(-5-\left(\left(\dfrac{17}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{280}{3}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{61}{9}\right)\\
&=&\left(\left(-\dfrac{73}{2}\right)\sqrt{7}+\dfrac{62}{63}\right)\left(\left(\dfrac{1090}{9}\right)\sqrt{7}-\dfrac{5075}{72}\right)\\
&=&\left(-\dfrac{39785}{9}\right)\sqrt{49}+\left(\dfrac{24421205}{9072}\right)\sqrt{7}-\dfrac{22475}{324}\\
&=&-\dfrac{10048295}{324}+\left(\dfrac{24421205}{9072}\right)\sqrt{7}\\
\end{eqnarray*}