L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{8}{9}\right)\sqrt{175}\) et \( Y=\left(\left(-\dfrac{44}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{63}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{28}\right)+\left(-\dfrac{21}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\dfrac{69}{8}\right)\sqrt{63}+3+\left(\dfrac{23}{3}\right)\sqrt{28}\) . 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{8}{9}\right)\sqrt{175}\right)+\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{63}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{28}\right)+\left(-\dfrac{21}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\dfrac{69}{8}\right)\sqrt{63}+3+\left(\dfrac{23}{3}\right)\sqrt{28}\right)\\
&=&\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)+\left(\left(\left(-\dfrac{88}{7}\right)\sqrt{7}\right)-\left(\left(66\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{2}\right)\sqrt{7}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{7}\right)+\left(-\dfrac{105}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(\dfrac{207}{8}\right)\sqrt{7}+3+\left(\dfrac{46}{3}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{40}{9}\right)\sqrt{7}+\left(\left(-\dfrac{88}{7}\right)\sqrt{7}\right)-\left(\left(66\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{2}\right)\sqrt{7}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{7}\right)+\left(-\dfrac{105}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(\dfrac{207}{8}\right)\sqrt{7}+3+\left(\dfrac{46}{3}\right)\sqrt{7}\\
&=&\left(-\dfrac{29611}{504}\right)\sqrt{7}+\dfrac{97}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{8}{9}\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{63}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{28}\right)+\left(-\dfrac{21}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\dfrac{69}{8}\right)\sqrt{63}+3+\left(\dfrac{23}{3}\right)\sqrt{28}\right)\\
&=&\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{88}{7}\right)\sqrt{7}\right)-\left(\left(66\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{2}\right)\sqrt{7}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{7}\right)+\left(-\dfrac{105}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(\dfrac{207}{8}\right)\sqrt{7}+3+\left(\dfrac{46}{3}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)-\left(\left(-\dfrac{3539}{56}\right)\sqrt{7}+\dfrac{97}{6}\right)\\
&=&\left(\dfrac{40}{9}\right)\sqrt{7}+\left(\dfrac{3539}{56}\right)\sqrt{7}-\dfrac{97}{6}\\
&=&\left(\dfrac{34091}{504}\right)\sqrt{7}-\dfrac{97}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{8}{9}\right)\sqrt{175}\right)\times\left(\left(\left(-\dfrac{44}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{175}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{63}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{28}\right)+\left(-\dfrac{21}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\dfrac{69}{8}\right)\sqrt{63}+3+\left(\dfrac{23}{3}\right)\sqrt{28}\right)\\
&=&\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)\times\left(\left(\left(-\dfrac{88}{7}\right)\sqrt{7}\right)-\left(\left(66\right)\sqrt{7}\right)-\left(\left(-\dfrac{57}{2}\right)\sqrt{7}\right)+\dfrac{79}{6}+\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{7}\right)+\left(-\dfrac{105}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}+\left(\dfrac{207}{8}\right)\sqrt{7}+3+\left(\dfrac{46}{3}\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{40}{9}\right)\sqrt{7}\right)\left(\left(-\dfrac{3539}{56}\right)\sqrt{7}+\dfrac{97}{6}\right)\\
&=&\left(-\dfrac{17695}{63}\right)\sqrt{49}+\left(\dfrac{1940}{27}\right)\sqrt{7}\\
&=&-\dfrac{17695}{9}+\left(\dfrac{1940}{27}\right)\sqrt{7}\\
\end{eqnarray*}