L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{66}{7}\right)\sqrt{75}+\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{12}\right)+\left(4\right)\sqrt{12}+\left(5\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{12}\) et \( Y=\left(\dfrac{29}{2}\right)\sqrt{9}\) . 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{66}{7}\right)\sqrt{75}+\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{12}\right)+\left(4\right)\sqrt{12}+\left(5\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{12}\right)+\left(\left(\dfrac{29}{2}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{330}{7}\right)\sqrt{3}+\dfrac{3}{5}-\left(\left(-\dfrac{130}{9}\right)\sqrt{3}\right)+50-\dfrac{3}{5}-\left(\left(27\right)\sqrt{3}\right)+\left(8\right)\sqrt{3}+\left(15\right)\sqrt{3}+\left(-5\right)\sqrt{3}\right)+\left(\dfrac{87}{2}\right)\\
&=&\left(\dfrac{330}{7}\right)\sqrt{3}+\dfrac{3}{5}-\left(\left(-\dfrac{130}{9}\right)\sqrt{3}\right)+50-\dfrac{3}{5}-\left(\left(27\right)\sqrt{3}\right)+\left(8\right)\sqrt{3}+\left(15\right)\sqrt{3}+\left(-5\right)\sqrt{3}+\dfrac{87}{2}\\
&=&\left(\dfrac{3313}{63}\right)\sqrt{3}+\dfrac{187}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{66}{7}\right)\sqrt{75}+\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{12}\right)+\left(4\right)\sqrt{12}+\left(5\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{330}{7}\right)\sqrt{3}+\dfrac{3}{5}-\left(\left(-\dfrac{130}{9}\right)\sqrt{3}\right)+50-\dfrac{3}{5}-\left(\left(27\right)\sqrt{3}\right)+\left(8\right)\sqrt{3}+\left(15\right)\sqrt{3}+\left(-5\right)\sqrt{3}\right)-\left(\dfrac{87}{2}\right)\\
&=&\left(\left(\dfrac{3313}{63}\right)\sqrt{3}+50\right)-\left(\dfrac{87}{2}\right)\\
&=&\left(\dfrac{3313}{63}\right)\sqrt{3}+50+-\dfrac{87}{2}\\
&=&\left(\dfrac{3313}{63}\right)\sqrt{3}+\dfrac{13}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{66}{7}\right)\sqrt{75}+\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{65}{9}\right)\sqrt{12}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{1}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{12}\right)+\left(4\right)\sqrt{12}+\left(5\right)\sqrt{27}+\left(-\dfrac{5}{2}\right)\sqrt{12}\right)\times\left(\left(\dfrac{29}{2}\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{330}{7}\right)\sqrt{3}+\dfrac{3}{5}-\left(\left(-\dfrac{130}{9}\right)\sqrt{3}\right)+50-\dfrac{3}{5}-\left(\left(27\right)\sqrt{3}\right)+\left(8\right)\sqrt{3}+\left(15\right)\sqrt{3}+\left(-5\right)\sqrt{3}\right)\times\left(\dfrac{87}{2}\right)\\
&=&\left(\left(\dfrac{3313}{63}\right)\sqrt{3}+50\right)\left(\dfrac{87}{2}\right)\\
&=&\left(\dfrac{96077}{42}\right)\sqrt{3}+2175\\
&=&\left(\dfrac{96077}{42}\right)\sqrt{3}+2175\\
\end{eqnarray*}