L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)-\left(-4-\left(\left(\dfrac{11}{5}\right)\sqrt{20}\right)\right)-\left(\left(7\right)\sqrt{20}\right)\) et \( Y=\left(\left(-\dfrac{59}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{20}\right)+\left(\dfrac{43}{6}\right)\sqrt{20}-\dfrac{10}{3}+\left(\dfrac{1}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{20}\) . 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{69}{4}\right)\sqrt{45}\right)-\left(-4-\left(\left(\dfrac{11}{5}\right)\sqrt{20}\right)\right)-\left(\left(7\right)\sqrt{20}\right)\right)+\left(\left(\left(-\dfrac{59}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{20}\right)+\left(\dfrac{43}{6}\right)\sqrt{20}-\dfrac{10}{3}+\left(\dfrac{1}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)-\left(-4-\left(\left(\dfrac{22}{5}\right)\sqrt{5}\right)\right)-\left(\left(14\right)\sqrt{5}\right)\right)+\left(\left(\left(-\dfrac{59}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{5}\right)+\left(\dfrac{43}{3}\right)\sqrt{5}-\dfrac{10}{3}+\left(\dfrac{5}{9}\right)\sqrt{5}+\left(-6\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)-\left(-4-\left(\left(\dfrac{22}{5}\right)\sqrt{5}\right)\right)-\left(\left(14\right)\sqrt{5}\right)+\left(\left(-\dfrac{59}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{5}\right)+\left(\dfrac{43}{3}\right)\sqrt{5}-\dfrac{10}{3}+\left(\dfrac{5}{9}\right)\sqrt{5}+\left(-6\right)\sqrt{5}\\
&=&\left(-\dfrac{2672}{45}\right)\sqrt{5}+\dfrac{2}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)-\left(-4-\left(\left(\dfrac{11}{5}\right)\sqrt{20}\right)\right)-\left(\left(7\right)\sqrt{20}\right)\right)-\left(\left(\left(-\dfrac{59}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{20}\right)+\left(\dfrac{43}{6}\right)\sqrt{20}-\dfrac{10}{3}+\left(\dfrac{1}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)-\left(-4-\left(\left(\dfrac{22}{5}\right)\sqrt{5}\right)\right)-\left(\left(14\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{59}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{5}\right)+\left(\dfrac{43}{3}\right)\sqrt{5}-\dfrac{10}{3}+\left(\dfrac{5}{9}\right)\sqrt{5}+\left(-6\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{1227}{20}\right)\sqrt{5}+4\right)-\left(\left(\dfrac{71}{36}\right)\sqrt{5}-\dfrac{10}{3}\right)\\
&=&\left(-\dfrac{1227}{20}\right)\sqrt{5}+4+\left(-\dfrac{71}{36}\right)\sqrt{5}+\dfrac{10}{3}\\
&=&\left(-\dfrac{5699}{90}\right)\sqrt{5}+\dfrac{22}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{69}{4}\right)\sqrt{45}\right)-\left(-4-\left(\left(\dfrac{11}{5}\right)\sqrt{20}\right)\right)-\left(\left(7\right)\sqrt{20}\right)\right)\times\left(\left(\left(-\dfrac{59}{6}\right)\sqrt{20}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{20}\right)-\left(\left(-\dfrac{31}{4}\right)\sqrt{20}\right)+\left(\dfrac{43}{6}\right)\sqrt{20}-\dfrac{10}{3}+\left(\dfrac{1}{9}\right)\sqrt{125}+\left(-3\right)\sqrt{20}\right)\\
&=&\left(\left(\left(-\dfrac{207}{4}\right)\sqrt{5}\right)-\left(-4-\left(\left(\dfrac{22}{5}\right)\sqrt{5}\right)\right)-\left(\left(14\right)\sqrt{5}\right)\right)\times\left(\left(\left(-\dfrac{59}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{31}{2}\right)\sqrt{5}\right)+\left(\dfrac{43}{3}\right)\sqrt{5}-\dfrac{10}{3}+\left(\dfrac{5}{9}\right)\sqrt{5}+\left(-6\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{1227}{20}\right)\sqrt{5}+4\right)\left(\left(\dfrac{71}{36}\right)\sqrt{5}-\dfrac{10}{3}\right)\\
&=&\left(-\dfrac{29039}{240}\right)\sqrt{25}+\left(\dfrac{3823}{18}\right)\sqrt{5}-\dfrac{40}{3}\\
&=&-\dfrac{9893}{16}+\left(\dfrac{3823}{18}\right)\sqrt{5}\\
\end{eqnarray*}