L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{67}{6}+\dfrac{76}{3}\) et \( Y=\left(-5\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{27}+\dfrac{3}{5}+\left(\dfrac{20}{7}\right)\sqrt{75}+\left(\left(\dfrac{21}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{64}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(-\dfrac{73}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{75}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{67}{6}+\dfrac{76}{3}\right)+\left(\left(-5\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{27}+\dfrac{3}{5}+\left(\dfrac{20}{7}\right)\sqrt{75}+\left(\left(\dfrac{21}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{64}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(-\dfrac{73}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{75}\right)\right)\\
&=&\left(\dfrac{67}{6}+\dfrac{76}{3}\right)+\left(-15+\left(\dfrac{33}{2}\right)\sqrt{3}+\dfrac{3}{5}+\left(\dfrac{100}{7}\right)\sqrt{3}+\left(\left(\dfrac{105}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{128}{7}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\left(\left(-\dfrac{365}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{3}\right)\right)\\
&=&\dfrac{67}{6}+\dfrac{76}{3}-15+\left(\dfrac{33}{2}\right)\sqrt{3}+\dfrac{3}{5}+\left(\dfrac{100}{7}\right)\sqrt{3}+\left(\left(\dfrac{105}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{128}{7}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\left(\left(-\dfrac{365}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{3}\right)\\
&=&\dfrac{221}{10}+\left(\dfrac{5377}{28}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{67}{6}+\dfrac{76}{3}\right)-\left(\left(-5\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{27}+\dfrac{3}{5}+\left(\dfrac{20}{7}\right)\sqrt{75}+\left(\left(\dfrac{21}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{64}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(-\dfrac{73}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{75}\right)\right)\\
&=&\left(\dfrac{67}{6}+\dfrac{76}{3}\right)-\left(-15+\left(\dfrac{33}{2}\right)\sqrt{3}+\dfrac{3}{5}+\left(\dfrac{100}{7}\right)\sqrt{3}+\left(\left(\dfrac{105}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{128}{7}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\left(\left(-\dfrac{365}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{73}{2}\right)-\left(-\dfrac{72}{5}+\left(\dfrac{5377}{28}\right)\sqrt{3}\right)\\
&=&\dfrac{73}{2}+\dfrac{72}{5}+\left(-\dfrac{5377}{28}\right)\sqrt{3}\\
&=&\dfrac{509}{10}+\left(-\dfrac{5377}{28}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{67}{6}+\dfrac{76}{3}\right)\times\left(\left(-5\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{27}+\dfrac{3}{5}+\left(\dfrac{20}{7}\right)\sqrt{75}+\left(\left(\dfrac{21}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{64}{7}\right)\sqrt{12}\right)-\left(\left(-7\right)\sqrt{27}\right)-\left(\left(-\dfrac{73}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{75}\right)\right)\\
&=&\left(\dfrac{67}{6}+\dfrac{76}{3}\right)\times\left(-15+\left(\dfrac{33}{2}\right)\sqrt{3}+\dfrac{3}{5}+\left(\dfrac{100}{7}\right)\sqrt{3}+\left(\left(\dfrac{105}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{128}{7}\right)\sqrt{3}\right)-\left(\left(-21\right)\sqrt{3}\right)-\left(\left(-\dfrac{365}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{305}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\dfrac{73}{2}\right)\left(-\dfrac{72}{5}+\left(\dfrac{5377}{28}\right)\sqrt{3}\right)\\
&=&-\dfrac{2628}{5}+\left(\dfrac{392521}{56}\right)\sqrt{3}\\
&=&-\dfrac{2628}{5}+\left(\dfrac{392521}{56}\right)\sqrt{3}\\
\end{eqnarray*}