L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-9\right)\sqrt{12}\) et \( Y=\left(\left(-\dfrac{71}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{32}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{27}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-9\right)\sqrt{12}\right)+\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{32}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{27}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(-18\right)\sqrt{3}\right)+\left(\left(\left(-\dfrac{355}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{160}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{132}{5}\right)\\
&=&\left(-18\right)\sqrt{3}+\left(\left(-\dfrac{355}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{160}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{132}{5}\\
&=&\left(-\dfrac{1985}{18}\right)\sqrt{3}-\dfrac{132}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-9\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{32}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{27}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(-18\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{355}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{160}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{132}{5}\right)\\
&=&\left(\left(-18\right)\sqrt{3}\right)-\left(\left(-\dfrac{1661}{18}\right)\sqrt{3}-\dfrac{132}{5}\right)\\
&=&\left(-18\right)\sqrt{3}+\left(\dfrac{1661}{18}\right)\sqrt{3}+\dfrac{132}{5}\\
&=&\left(\dfrac{1337}{18}\right)\sqrt{3}+\dfrac{132}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-9\right)\sqrt{12}\right)\times\left(\left(\left(-\dfrac{71}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{32}{9}\right)\sqrt{75}\right)-\left(\left(\dfrac{9}{4}\right)\sqrt{27}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(\dfrac{44}{5}\right)\sqrt{9}\right)\right)\\
&=&\left(\left(-18\right)\sqrt{3}\right)\times\left(\left(\left(-\dfrac{355}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{160}{9}\right)\sqrt{3}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\dfrac{132}{5}\right)\\
&=&\left(\left(-18\right)\sqrt{3}\right)\left(\left(-\dfrac{1661}{18}\right)\sqrt{3}-\dfrac{132}{5}\right)\\
&=&\left(1661\right)\sqrt{9}+\left(\dfrac{2376}{5}\right)\sqrt{3}\\
&=&4983+\left(\dfrac{2376}{5}\right)\sqrt{3}\\
\end{eqnarray*}