L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)+\left(\dfrac{32}{3}\right)\sqrt{27}+\left(-1\right)\sqrt{27}\) et \( Y=\left(\dfrac{61}{9}\right)\sqrt{9}+\left(-1\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{39}{4}\right)\sqrt{75}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)+\left(\dfrac{32}{3}\right)\sqrt{27}+\left(-1\right)\sqrt{27}\right)+\left(\left(\dfrac{61}{9}\right)\sqrt{9}+\left(-1\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{39}{4}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(21\right)\sqrt{3}\right)-\left(\left(56\right)\sqrt{3}\right)+\left(32\right)\sqrt{3}+\left(-3\right)\sqrt{3}\right)+\left(\dfrac{61}{3}+\left(-5\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}\right)\\
&=&\left(\left(21\right)\sqrt{3}\right)-\left(\left(56\right)\sqrt{3}\right)+\left(32\right)\sqrt{3}+\left(-3\right)\sqrt{3}+\dfrac{61}{3}+\left(-5\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}\\
&=&\left(79\right)\sqrt{3}+\dfrac{61}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)+\left(\dfrac{32}{3}\right)\sqrt{27}+\left(-1\right)\sqrt{27}\right)-\left(\left(\dfrac{61}{9}\right)\sqrt{9}+\left(-1\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{39}{4}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(21\right)\sqrt{3}\right)-\left(\left(56\right)\sqrt{3}\right)+\left(32\right)\sqrt{3}+\left(-3\right)\sqrt{3}\right)-\left(\dfrac{61}{3}+\left(-5\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}\right)\\
&=&\left(\left(-6\right)\sqrt{3}\right)-\left(\dfrac{61}{3}+\left(85\right)\sqrt{3}\right)\\
&=&\left(-6\right)\sqrt{3}+-\dfrac{61}{3}+\left(-85\right)\sqrt{3}\\
&=&\left(-91\right)\sqrt{3}-\dfrac{61}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{21}{2}\right)\sqrt{12}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)+\left(\dfrac{32}{3}\right)\sqrt{27}+\left(-1\right)\sqrt{27}\right)\times\left(\left(\dfrac{61}{9}\right)\sqrt{9}+\left(-1\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{33}{8}\right)\sqrt{75}+\left(\dfrac{39}{4}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(21\right)\sqrt{3}\right)-\left(\left(56\right)\sqrt{3}\right)+\left(32\right)\sqrt{3}+\left(-3\right)\sqrt{3}\right)\times\left(\dfrac{61}{3}+\left(-5\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{165}{8}\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}\right)\\
&=&\left(\left(-6\right)\sqrt{3}\right)\left(\dfrac{61}{3}+\left(85\right)\sqrt{3}\right)\\
&=&\left(-122\right)\sqrt{3}+\left(-510\right)\sqrt{9}\\
&=&\left(-122\right)\sqrt{3}-1530\\
\end{eqnarray*}