L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-2\right)\sqrt{27}+\left(1\right)\sqrt{75}+\left(\dfrac{70}{3}\right)\sqrt{12}+\left(\dfrac{73}{8}\right)\sqrt{75}+\left(-3\right)\sqrt{9}\right)-\left(-\dfrac{8}{5}+\left(\dfrac{65}{9}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{59}{2}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{61}{9}\right)\sqrt{27}\right)\) et \( Y=\dfrac{28}{3}-\left(\left(-\dfrac{79}{4}\right)\sqrt{27}\right)-9-\left(\left(7\right)\sqrt{12}\right)-\dfrac{51}{7}+\left(-5\right)\sqrt{9}\) . 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(-2\right)\sqrt{27}+\left(1\right)\sqrt{75}+\left(\dfrac{70}{3}\right)\sqrt{12}+\left(\dfrac{73}{8}\right)\sqrt{75}+\left(-3\right)\sqrt{9}\right)-\left(-\dfrac{8}{5}+\left(\dfrac{65}{9}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{59}{2}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{61}{9}\right)\sqrt{27}\right)\right)+\left(\dfrac{28}{3}-\left(\left(-\dfrac{79}{4}\right)\sqrt{27}\right)-9-\left(\left(7\right)\sqrt{12}\right)-\dfrac{51}{7}+\left(-5\right)\sqrt{9}\right)\\
&=&\left(\left(\left(-6\right)\sqrt{3}+\left(5\right)\sqrt{3}+\left(\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{365}{8}\right)\sqrt{3}-9\right)-\left(-\dfrac{8}{5}+\left(\dfrac{325}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{115}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{295}{2}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{3}\right)\right)+\left(\dfrac{28}{3}-\left(\left(-\dfrac{237}{4}\right)\sqrt{3}\right)-9-\left(\left(14\right)\sqrt{3}\right)-\dfrac{51}{7}-15\right)\\
&=&\left(\left(-6\right)\sqrt{3}+\left(5\right)\sqrt{3}+\left(\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{365}{8}\right)\sqrt{3}-9\right)-\left(-\dfrac{8}{5}+\left(\dfrac{325}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{115}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{295}{2}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{3}\right)+\dfrac{28}{3}-\left(\left(-\dfrac{237}{4}\right)\sqrt{3}\right)-9-\left(\left(14\right)\sqrt{3}\right)-\dfrac{51}{7}-15\\
&=&\left(-\dfrac{42251}{504}\right)\sqrt{3}-\dfrac{3082}{105}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-2\right)\sqrt{27}+\left(1\right)\sqrt{75}+\left(\dfrac{70}{3}\right)\sqrt{12}+\left(\dfrac{73}{8}\right)\sqrt{75}+\left(-3\right)\sqrt{9}\right)-\left(-\dfrac{8}{5}+\left(\dfrac{65}{9}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{59}{2}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{61}{9}\right)\sqrt{27}\right)\right)-\left(\dfrac{28}{3}-\left(\left(-\dfrac{79}{4}\right)\sqrt{27}\right)-9-\left(\left(7\right)\sqrt{12}\right)-\dfrac{51}{7}+\left(-5\right)\sqrt{9}\right)\\
&=&\left(\left(\left(-6\right)\sqrt{3}+\left(5\right)\sqrt{3}+\left(\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{365}{8}\right)\sqrt{3}-9\right)-\left(-\dfrac{8}{5}+\left(\dfrac{325}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{115}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{295}{2}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{3}\right)\right)-\left(\dfrac{28}{3}-\left(\left(-\dfrac{237}{4}\right)\sqrt{3}\right)-9-\left(\left(14\right)\sqrt{3}\right)-\dfrac{51}{7}-15\right)\\
&=&\left(\left(-\dfrac{65057}{504}\right)\sqrt{3}-\dfrac{37}{5}\right)-\left(-\dfrac{461}{21}+\left(\dfrac{181}{4}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{65057}{504}\right)\sqrt{3}-\dfrac{37}{5}+\dfrac{461}{21}+\left(-\dfrac{181}{4}\right)\sqrt{3}\\
&=&\left(-\dfrac{87863}{504}\right)\sqrt{3}+\dfrac{1528}{105}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-2\right)\sqrt{27}+\left(1\right)\sqrt{75}+\left(\dfrac{70}{3}\right)\sqrt{12}+\left(\dfrac{73}{8}\right)\sqrt{75}+\left(-3\right)\sqrt{9}\right)-\left(-\dfrac{8}{5}+\left(\dfrac{65}{9}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{59}{2}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{61}{9}\right)\sqrt{27}\right)\right)\times\left(\dfrac{28}{3}-\left(\left(-\dfrac{79}{4}\right)\sqrt{27}\right)-9-\left(\left(7\right)\sqrt{12}\right)-\dfrac{51}{7}+\left(-5\right)\sqrt{9}\right)\\
&=&\left(\left(\left(-6\right)\sqrt{3}+\left(5\right)\sqrt{3}+\left(\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{365}{8}\right)\sqrt{3}-9\right)-\left(-\dfrac{8}{5}+\left(\dfrac{325}{9}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{115}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{295}{2}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{61}{3}\right)\sqrt{3}\right)\right)\times\left(\dfrac{28}{3}-\left(\left(-\dfrac{237}{4}\right)\sqrt{3}\right)-9-\left(\left(14\right)\sqrt{3}\right)-\dfrac{51}{7}-15\right)\\
&=&\left(\left(-\dfrac{65057}{504}\right)\sqrt{3}-\dfrac{37}{5}\right)\left(-\dfrac{461}{21}+\left(\dfrac{181}{4}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{132236123}{52920}\right)\sqrt{3}+\left(-\dfrac{11775317}{2016}\right)\sqrt{9}+\dfrac{17057}{105}\\
&=&\left(\dfrac{132236123}{52920}\right)\sqrt{3}-\dfrac{19443587}{1120}\\
\end{eqnarray*}