L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-\dfrac{61}{5}\right)\sqrt{28}\) et \( Y=\left(\left(-\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(4\right)\sqrt{49}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{49}\right)+\left(\dfrac{7}{6}\right)\sqrt{28}+\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{18}{5}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{28}\right)+\left(\left(\dfrac{35}{4}\right)\sqrt{63}\right)-\dfrac{5}{3}\) . 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{61}{5}\right)\sqrt{28}\right)+\left(\left(\left(-\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(4\right)\sqrt{49}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{49}\right)+\left(\dfrac{7}{6}\right)\sqrt{28}+\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{18}{5}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{28}\right)+\left(\left(\dfrac{35}{4}\right)\sqrt{63}\right)-\dfrac{5}{3}\right)\\
&=&\left(\left(-\dfrac{122}{5}\right)\sqrt{7}\right)+\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{7}\right)-28+\dfrac{56}{3}+\left(\dfrac{7}{3}\right)\sqrt{7}+\left(\left(10\right)\sqrt{7}\right)+\dfrac{126}{5}-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{134}{5}\right)\sqrt{7}\right)+\left(\left(\dfrac{105}{4}\right)\sqrt{7}\right)-\dfrac{5}{3}\right)\\
&=&\left(-\dfrac{122}{5}\right)\sqrt{7}+\left(\left(-\dfrac{56}{9}\right)\sqrt{7}\right)-28+\dfrac{56}{3}+\left(\dfrac{7}{3}\right)\sqrt{7}+\left(\left(10\right)\sqrt{7}\right)+\dfrac{126}{5}-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{134}{5}\right)\sqrt{7}\right)+\left(\left(\dfrac{105}{4}\right)\sqrt{7}\right)-\dfrac{5}{3}\\
&=&\left(\dfrac{4457}{180}\right)\sqrt{7}+\dfrac{71}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{61}{5}\right)\sqrt{28}\right)-\left(\left(\left(-\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(4\right)\sqrt{49}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{49}\right)+\left(\dfrac{7}{6}\right)\sqrt{28}+\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{18}{5}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{28}\right)+\left(\left(\dfrac{35}{4}\right)\sqrt{63}\right)-\dfrac{5}{3}\right)\\
&=&\left(\left(-\dfrac{122}{5}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{7}\right)-28+\dfrac{56}{3}+\left(\dfrac{7}{3}\right)\sqrt{7}+\left(\left(10\right)\sqrt{7}\right)+\dfrac{126}{5}-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{134}{5}\right)\sqrt{7}\right)+\left(\left(\dfrac{105}{4}\right)\sqrt{7}\right)-\dfrac{5}{3}\right)\\
&=&\left(\left(-\dfrac{122}{5}\right)\sqrt{7}\right)-\left(\left(\dfrac{8849}{180}\right)\sqrt{7}+\dfrac{71}{5}\right)\\
&=&\left(-\dfrac{122}{5}\right)\sqrt{7}+\left(-\dfrac{8849}{180}\right)\sqrt{7}-\dfrac{71}{5}\\
&=&\left(-\dfrac{13241}{180}\right)\sqrt{7}-\dfrac{71}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{61}{5}\right)\sqrt{28}\right)\times\left(\left(\left(-\dfrac{28}{9}\right)\sqrt{28}\right)-\left(\left(4\right)\sqrt{49}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{49}\right)+\left(\dfrac{7}{6}\right)\sqrt{28}+\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{18}{5}\right)\sqrt{49}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{67}{5}\right)\sqrt{28}\right)+\left(\left(\dfrac{35}{4}\right)\sqrt{63}\right)-\dfrac{5}{3}\right)\\
&=&\left(\left(-\dfrac{122}{5}\right)\sqrt{7}\right)\times\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{7}\right)-28+\dfrac{56}{3}+\left(\dfrac{7}{3}\right)\sqrt{7}+\left(\left(10\right)\sqrt{7}\right)+\dfrac{126}{5}-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{134}{5}\right)\sqrt{7}\right)+\left(\left(\dfrac{105}{4}\right)\sqrt{7}\right)-\dfrac{5}{3}\right)\\
&=&\left(\left(-\dfrac{122}{5}\right)\sqrt{7}\right)\left(\left(\dfrac{8849}{180}\right)\sqrt{7}+\dfrac{71}{5}\right)\\
&=&\left(-\dfrac{539789}{450}\right)\sqrt{49}+\left(-\dfrac{8662}{25}\right)\sqrt{7}\\
&=&-\dfrac{3778523}{450}+\left(-\dfrac{8662}{25}\right)\sqrt{7}\\
\end{eqnarray*}