L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-\dfrac{59}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{9}\right)\) et \( Y=\left(\left(\left(-5\right)\sqrt{75}\right)-5\right)-\left(\left(-6\right)\sqrt{27}+\left(-\dfrac{65}{8}\right)\sqrt{9}+\left(\dfrac{63}{8}\right)\sqrt{27}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-\dfrac{59}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{9}\right)\right)+\left(\left(\left(\left(-5\right)\sqrt{75}\right)-5\right)-\left(\left(-6\right)\sqrt{27}+\left(-\dfrac{65}{8}\right)\sqrt{9}+\left(\dfrac{63}{8}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-\dfrac{295}{7}\right)\sqrt{3}\right)+\dfrac{171}{4}\right)+\left(\left(\left(\left(-25\right)\sqrt{3}\right)-5\right)-\left(\left(-18\right)\sqrt{3}-\dfrac{195}{8}+\left(\dfrac{189}{8}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-\dfrac{295}{7}\right)\sqrt{3}\right)+\dfrac{171}{4}+\left(\left(\left(-25\right)\sqrt{3}\right)-5\right)-\left(\left(-18\right)\sqrt{3}-\dfrac{195}{8}+\left(\dfrac{189}{8}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{4219}{56}\right)\sqrt{3}+\dfrac{497}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-\dfrac{59}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{9}\right)\right)-\left(\left(\left(\left(-5\right)\sqrt{75}\right)-5\right)-\left(\left(-6\right)\sqrt{27}+\left(-\dfrac{65}{8}\right)\sqrt{9}+\left(\dfrac{63}{8}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-\dfrac{295}{7}\right)\sqrt{3}\right)+\dfrac{171}{4}\right)-\left(\left(\left(\left(-25\right)\sqrt{3}\right)-5\right)-\left(\left(-18\right)\sqrt{3}-\dfrac{195}{8}+\left(\dfrac{189}{8}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{313}{7}\right)\sqrt{3}+\dfrac{171}{4}\right)-\left(\left(-\dfrac{245}{8}\right)\sqrt{3}+\dfrac{155}{8}\right)\\
&=&\left(-\dfrac{313}{7}\right)\sqrt{3}+\dfrac{171}{4}+\left(\dfrac{245}{8}\right)\sqrt{3}-\dfrac{155}{8}\\
&=&\left(-\dfrac{789}{56}\right)\sqrt{3}+\dfrac{187}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{6}{7}\right)\sqrt{27}+\left(-\dfrac{59}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{57}{4}\right)\sqrt{9}\right)\right)\times\left(\left(\left(\left(-5\right)\sqrt{75}\right)-5\right)-\left(\left(-6\right)\sqrt{27}+\left(-\dfrac{65}{8}\right)\sqrt{9}+\left(\dfrac{63}{8}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(-\dfrac{18}{7}\right)\sqrt{3}+\left(-\dfrac{295}{7}\right)\sqrt{3}\right)+\dfrac{171}{4}\right)\times\left(\left(\left(\left(-25\right)\sqrt{3}\right)-5\right)-\left(\left(-18\right)\sqrt{3}-\dfrac{195}{8}+\left(\dfrac{189}{8}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{313}{7}\right)\sqrt{3}+\dfrac{171}{4}\right)\left(\left(-\dfrac{245}{8}\right)\sqrt{3}+\dfrac{155}{8}\right)\\
&=&\left(\dfrac{10955}{8}\right)\sqrt{9}+\left(-\dfrac{487325}{224}\right)\sqrt{3}+\dfrac{26505}{32}\\
&=&\dfrac{157965}{32}+\left(-\dfrac{487325}{224}\right)\sqrt{3}\\
\end{eqnarray*}