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.
Soit \( X=\left(-\dfrac{77}{5}\right)\sqrt{12}+\left(\left(-\dfrac{27}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{75}\right)+\left(\left(-\dfrac{14}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{27}\right)\) et \( Y=\left(-\dfrac{35}{3}\right)\sqrt{27}+\left(-\dfrac{78}{7}\right)\sqrt{75}+\left(-\dfrac{22}{3}\right)\sqrt{27}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{77}{5}\right)\sqrt{12}+\left(\left(-\dfrac{27}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{75}\right)+\left(\left(-\dfrac{14}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{27}\right)\right)+\left(\left(-\dfrac{35}{3}\right)\sqrt{27}+\left(-\dfrac{78}{7}\right)\sqrt{75}+\left(-\dfrac{22}{3}\right)\sqrt{27}\right)\\
&=&\left(\left(-\dfrac{154}{5}\right)\sqrt{3}+\left(\left(-\dfrac{54}{7}\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)-\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\dfrac{42}{5}-\left(\left(-18\right)\sqrt{3}\right)\right)+\left(\left(-35\right)\sqrt{3}+\left(-\dfrac{390}{7}\right)\sqrt{3}+\left(-22\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{154}{5}\right)\sqrt{3}+\left(\left(-\dfrac{54}{7}\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)-\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\dfrac{42}{5}-\left(\left(-18\right)\sqrt{3}\right)+\left(-35\right)\sqrt{3}+\left(-\dfrac{390}{7}\right)\sqrt{3}+\left(-22\right)\sqrt{3}\\
&=&\left(-\dfrac{42107}{315}\right)\sqrt{3}-\dfrac{42}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{77}{5}\right)\sqrt{12}+\left(\left(-\dfrac{27}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{75}\right)+\left(\left(-\dfrac{14}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{27}+\left(-\dfrac{78}{7}\right)\sqrt{75}+\left(-\dfrac{22}{3}\right)\sqrt{27}\right)\\
&=&\left(\left(-\dfrac{154}{5}\right)\sqrt{3}+\left(\left(-\dfrac{54}{7}\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)-\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\dfrac{42}{5}-\left(\left(-18\right)\sqrt{3}\right)\right)-\left(\left(-35\right)\sqrt{3}+\left(-\dfrac{390}{7}\right)\sqrt{3}+\left(-22\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{6602}{315}\right)\sqrt{3}-\dfrac{42}{5}\right)-\left(\left(-\dfrac{789}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{6602}{315}\right)\sqrt{3}-\dfrac{42}{5}+\left(\dfrac{789}{7}\right)\sqrt{3}\\
&=&\left(\dfrac{4129}{45}\right)\sqrt{3}-\dfrac{42}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{77}{5}\right)\sqrt{12}+\left(\left(-\dfrac{27}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{12}\right)-\left(\left(\dfrac{26}{9}\right)\sqrt{75}\right)+\left(\left(-\dfrac{14}{5}\right)\sqrt{9}\right)-\left(\left(-6\right)\sqrt{27}\right)\right)\times\left(\left(-\dfrac{35}{3}\right)\sqrt{27}+\left(-\dfrac{78}{7}\right)\sqrt{75}+\left(-\dfrac{22}{3}\right)\sqrt{27}\right)\\
&=&\left(\left(-\dfrac{154}{5}\right)\sqrt{3}+\left(\left(-\dfrac{54}{7}\right)\sqrt{3}\right)-\left(\left(-14\right)\sqrt{3}\right)-\left(\left(\dfrac{130}{9}\right)\sqrt{3}\right)-\dfrac{42}{5}-\left(\left(-18\right)\sqrt{3}\right)\right)\times\left(\left(-35\right)\sqrt{3}+\left(-\dfrac{390}{7}\right)\sqrt{3}+\left(-22\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{6602}{315}\right)\sqrt{3}-\dfrac{42}{5}\right)\left(\left(-\dfrac{789}{7}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{1736326}{735}\right)\sqrt{9}+\left(\dfrac{4734}{5}\right)\sqrt{3}\\
&=&\dfrac{1736326}{245}+\left(\dfrac{4734}{5}\right)\sqrt{3}\\
\end{eqnarray*}