L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-6\right)\sqrt{12}\) et \( Y=\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{12}\right)\right)-\left(\left(-5\right)\sqrt{27}+\dfrac{7}{3}+\left(-8\right)\sqrt{75}+\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(-4\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{9}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-6\right)\sqrt{12}\right)+\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{12}\right)\right)-\left(\left(-5\right)\sqrt{27}+\dfrac{7}{3}+\left(-8\right)\sqrt{75}+\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(-4\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-12\right)\sqrt{3}\right)+\left(\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)-\left(\left(-8\right)\sqrt{3}\right)\right)-\left(\left(-15\right)\sqrt{3}+\dfrac{7}{3}+\left(-40\right)\sqrt{3}-\dfrac{63}{4}\right)-\left(-\dfrac{63}{4}-\left(\left(-20\right)\sqrt{3}\right)+\dfrac{153}{2}+\dfrac{45}{7}\right)\right)\\
&=&\left(-12\right)\sqrt{3}+\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)-\left(\left(-8\right)\sqrt{3}\right)\right)-\left(\left(-15\right)\sqrt{3}+\dfrac{7}{3}+\left(-40\right)\sqrt{3}-\dfrac{63}{4}\right)-\left(-\dfrac{63}{4}-\left(\left(-20\right)\sqrt{3}\right)+\dfrac{153}{2}+\dfrac{45}{7}\right)\\
&=&\left(-\dfrac{1019}{21}\right)\sqrt{3}-\dfrac{12077}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-6\right)\sqrt{12}\right)-\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{12}\right)\right)-\left(\left(-5\right)\sqrt{27}+\dfrac{7}{3}+\left(-8\right)\sqrt{75}+\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(-4\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-12\right)\sqrt{3}\right)-\left(\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)-\left(\left(-8\right)\sqrt{3}\right)\right)-\left(\left(-15\right)\sqrt{3}+\dfrac{7}{3}+\left(-40\right)\sqrt{3}-\dfrac{63}{4}\right)-\left(-\dfrac{63}{4}-\left(\left(-20\right)\sqrt{3}\right)+\dfrac{153}{2}+\dfrac{45}{7}\right)\right)\\
&=&\left(\left(-12\right)\sqrt{3}\right)-\left(\left(-\dfrac{767}{21}\right)\sqrt{3}-\dfrac{12077}{168}\right)\\
&=&\left(-12\right)\sqrt{3}+\left(\dfrac{767}{21}\right)\sqrt{3}+\dfrac{12077}{168}\\
&=&\left(\dfrac{515}{21}\right)\sqrt{3}+\dfrac{12077}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-6\right)\sqrt{12}\right)\times\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{16}{3}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{12}\right)\right)-\left(\left(-5\right)\sqrt{27}+\dfrac{7}{3}+\left(-8\right)\sqrt{75}+\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{21}{4}\right)\sqrt{9}\right)-\left(\left(-4\right)\sqrt{75}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{9}\right)-\left(\left(-\dfrac{15}{7}\right)\sqrt{9}\right)\right)\right)\\
&=&\left(\left(-12\right)\sqrt{3}\right)\times\left(\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)-\left(\dfrac{33}{4}+\dfrac{79}{8}-\left(\left(-\dfrac{32}{3}\right)\sqrt{3}\right)-\left(\left(-8\right)\sqrt{3}\right)\right)-\left(\left(-15\right)\sqrt{3}+\dfrac{7}{3}+\left(-40\right)\sqrt{3}-\dfrac{63}{4}\right)-\left(-\dfrac{63}{4}-\left(\left(-20\right)\sqrt{3}\right)+\dfrac{153}{2}+\dfrac{45}{7}\right)\right)\\
&=&\left(\left(-12\right)\sqrt{3}\right)\left(\left(-\dfrac{767}{21}\right)\sqrt{3}-\dfrac{12077}{168}\right)\\
&=&\left(\dfrac{3068}{7}\right)\sqrt{9}+\left(\dfrac{12077}{14}\right)\sqrt{3}\\
&=&\dfrac{9204}{7}+\left(\dfrac{12077}{14}\right)\sqrt{3}\\
\end{eqnarray*}