L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(2\right)\sqrt{12}+\left(\dfrac{66}{5}\right)\sqrt{75}-3+\left(-\dfrac{9}{2}\right)\sqrt{9}+\left(-\dfrac{27}{4}\right)\sqrt{9}+\left(-\dfrac{43}{8}\right)\sqrt{75}+\left(-2\right)\sqrt{9}+\dfrac{5}{2}+\left(-\dfrac{5}{6}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{12}+\dfrac{68}{9}\) et \( Y=\left(-\dfrac{55}{4}\right)\sqrt{75}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(2\right)\sqrt{12}+\left(\dfrac{66}{5}\right)\sqrt{75}-3+\left(-\dfrac{9}{2}\right)\sqrt{9}+\left(-\dfrac{27}{4}\right)\sqrt{9}+\left(-\dfrac{43}{8}\right)\sqrt{75}+\left(-2\right)\sqrt{9}+\dfrac{5}{2}+\left(-\dfrac{5}{6}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{12}+\dfrac{68}{9}\right)+\left(\left(-\dfrac{55}{4}\right)\sqrt{75}\right)\\
&=&\left(\left(4\right)\sqrt{3}+\left(66\right)\sqrt{3}-3-\dfrac{27}{2}-\dfrac{81}{4}+\left(-\dfrac{215}{8}\right)\sqrt{3}-6+\dfrac{5}{2}+\left(-\dfrac{5}{2}\right)\sqrt{3}+\left(-\dfrac{76}{5}\right)\sqrt{3}+\dfrac{68}{9}\right)+\left(\left(-\dfrac{275}{4}\right)\sqrt{3}\right)\\
&=&\left(4\right)\sqrt{3}+\left(66\right)\sqrt{3}-3-\dfrac{27}{2}-\dfrac{81}{4}+\left(-\dfrac{215}{8}\right)\sqrt{3}-6+\dfrac{5}{2}+\left(-\dfrac{5}{2}\right)\sqrt{3}+\left(-\dfrac{76}{5}\right)\sqrt{3}+\dfrac{68}{9}+\left(-\dfrac{275}{4}\right)\sqrt{3}\\
&=&\left(-\dfrac{1733}{40}\right)\sqrt{3}-\dfrac{1177}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(2\right)\sqrt{12}+\left(\dfrac{66}{5}\right)\sqrt{75}-3+\left(-\dfrac{9}{2}\right)\sqrt{9}+\left(-\dfrac{27}{4}\right)\sqrt{9}+\left(-\dfrac{43}{8}\right)\sqrt{75}+\left(-2\right)\sqrt{9}+\dfrac{5}{2}+\left(-\dfrac{5}{6}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{12}+\dfrac{68}{9}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{75}\right)\\
&=&\left(\left(4\right)\sqrt{3}+\left(66\right)\sqrt{3}-3-\dfrac{27}{2}-\dfrac{81}{4}+\left(-\dfrac{215}{8}\right)\sqrt{3}-6+\dfrac{5}{2}+\left(-\dfrac{5}{2}\right)\sqrt{3}+\left(-\dfrac{76}{5}\right)\sqrt{3}+\dfrac{68}{9}\right)-\left(\left(-\dfrac{275}{4}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{1017}{40}\right)\sqrt{3}-\dfrac{1177}{36}\right)-\left(\left(-\dfrac{275}{4}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{1017}{40}\right)\sqrt{3}-\dfrac{1177}{36}+\left(\dfrac{275}{4}\right)\sqrt{3}\\
&=&\left(\dfrac{3767}{40}\right)\sqrt{3}-\dfrac{1177}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(2\right)\sqrt{12}+\left(\dfrac{66}{5}\right)\sqrt{75}-3+\left(-\dfrac{9}{2}\right)\sqrt{9}+\left(-\dfrac{27}{4}\right)\sqrt{9}+\left(-\dfrac{43}{8}\right)\sqrt{75}+\left(-2\right)\sqrt{9}+\dfrac{5}{2}+\left(-\dfrac{5}{6}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{12}+\dfrac{68}{9}\right)\times\left(\left(-\dfrac{55}{4}\right)\sqrt{75}\right)\\
&=&\left(\left(4\right)\sqrt{3}+\left(66\right)\sqrt{3}-3-\dfrac{27}{2}-\dfrac{81}{4}+\left(-\dfrac{215}{8}\right)\sqrt{3}-6+\dfrac{5}{2}+\left(-\dfrac{5}{2}\right)\sqrt{3}+\left(-\dfrac{76}{5}\right)\sqrt{3}+\dfrac{68}{9}\right)\times\left(\left(-\dfrac{275}{4}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{1017}{40}\right)\sqrt{3}-\dfrac{1177}{36}\right)\left(\left(-\dfrac{275}{4}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{55935}{32}\right)\sqrt{9}+\left(\dfrac{323675}{144}\right)\sqrt{3}\\
&=&-\dfrac{167805}{32}+\left(\dfrac{323675}{144}\right)\sqrt{3}\\
\end{eqnarray*}