L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{48}{5}\right)\sqrt{9}\) et \( Y=\left(\left(\dfrac{32}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{12}+\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-4\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{27}+\left(-\dfrac{50}{3}\right)\sqrt{75}\right)\) . 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{48}{5}\right)\sqrt{9}\right)+\left(\left(\left(\dfrac{32}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{12}+\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-4\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{27}+\left(-\dfrac{50}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\dfrac{144}{5}\right)+\left(\dfrac{96}{5}-\left(\left(-\dfrac{30}{7}\right)\sqrt{3}+\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-8\right)\sqrt{3}\right)-\left(\left(3\right)\sqrt{3}+\left(-\dfrac{250}{3}\right)\sqrt{3}\right)\right)\\
&=&\dfrac{144}{5}+\dfrac{96}{5}-\left(\left(-\dfrac{30}{7}\right)\sqrt{3}+\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-8\right)\sqrt{3}\right)-\left(\left(3\right)\sqrt{3}+\left(-\dfrac{250}{3}\right)\sqrt{3}\right)\\
&=&48+\left(\dfrac{1999}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{48}{5}\right)\sqrt{9}\right)-\left(\left(\left(\dfrac{32}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{12}+\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-4\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{27}+\left(-\dfrac{50}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\dfrac{144}{5}\right)-\left(\dfrac{96}{5}-\left(\left(-\dfrac{30}{7}\right)\sqrt{3}+\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-8\right)\sqrt{3}\right)-\left(\left(3\right)\sqrt{3}+\left(-\dfrac{250}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{144}{5}\right)-\left(\dfrac{96}{5}+\left(\dfrac{1999}{21}\right)\sqrt{3}\right)\\
&=&\dfrac{144}{5}+-\dfrac{96}{5}+\left(-\dfrac{1999}{21}\right)\sqrt{3}\\
&=&\dfrac{48}{5}+\left(-\dfrac{1999}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{48}{5}\right)\sqrt{9}\right)\times\left(\left(\left(\dfrac{32}{5}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{12}+\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-4\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{27}+\left(-\dfrac{50}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\dfrac{144}{5}\right)\times\left(\dfrac{96}{5}-\left(\left(-\dfrac{30}{7}\right)\sqrt{3}+\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-8\right)\sqrt{3}\right)-\left(\left(3\right)\sqrt{3}+\left(-\dfrac{250}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{144}{5}\right)\left(\dfrac{96}{5}+\left(\dfrac{1999}{21}\right)\sqrt{3}\right)\\
&=&\dfrac{13824}{25}+\left(\dfrac{95952}{35}\right)\sqrt{3}\\
&=&\dfrac{13824}{25}+\left(\dfrac{95952}{35}\right)\sqrt{3}\\
\end{eqnarray*}