L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(-4\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{9}+\dfrac{60}{7}+\left(\left(2\right)\sqrt{9}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{73}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{80}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{78}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{75}\right)+6-\left(\left(-\dfrac{79}{4}\right)\sqrt{12}\right)\) et \( Y=\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{15}{4}\right)\sqrt{75}-\dfrac{13}{3}+\left(-\dfrac{71}{9}\right)\sqrt{12}+\left(-\dfrac{77}{4}\right)\sqrt{75}+\left(\dfrac{8}{3}\right)\sqrt{27}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-4\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{9}+\dfrac{60}{7}+\left(\left(2\right)\sqrt{9}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{73}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{80}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{78}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{75}\right)+6-\left(\left(-\dfrac{79}{4}\right)\sqrt{12}\right)\right)+\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{15}{4}\right)\sqrt{75}-\dfrac{13}{3}+\left(-\dfrac{71}{9}\right)\sqrt{12}+\left(-\dfrac{77}{4}\right)\sqrt{75}+\left(\dfrac{8}{3}\right)\sqrt{27}\right)\\
&=&\left(\left(-20\right)\sqrt{3}+2+\dfrac{60}{7}+6-\left(\left(-\dfrac{37}{3}\right)\sqrt{3}\right)-\dfrac{219}{4}-\left(\left(\dfrac{240}{7}\right)\sqrt{3}\right)+\left(\left(-78\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)+6-\left(\left(-\dfrac{79}{2}\right)\sqrt{3}\right)\right)+\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\left(\dfrac{75}{4}\right)\sqrt{3}-\dfrac{13}{3}+\left(-\dfrac{142}{9}\right)\sqrt{3}+\left(-\dfrac{385}{4}\right)\sqrt{3}+\left(8\right)\sqrt{3}\right)\\
&=&\left(-20\right)\sqrt{3}+2+\dfrac{60}{7}+6-\left(\left(-\dfrac{37}{3}\right)\sqrt{3}\right)-\dfrac{219}{4}-\left(\left(\dfrac{240}{7}\right)\sqrt{3}\right)+\left(\left(-78\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)+6-\left(\left(-\dfrac{79}{2}\right)\sqrt{3}\right)+\left(\dfrac{34}{9}\right)\sqrt{3}+\left(\dfrac{75}{4}\right)\sqrt{3}-\dfrac{13}{3}+\left(-\dfrac{142}{9}\right)\sqrt{3}+\left(-\dfrac{385}{4}\right)\sqrt{3}+\left(8\right)\sqrt{3}\\
&=&\left(-\dfrac{3443}{21}\right)\sqrt{3}-\dfrac{3067}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-4\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{9}+\dfrac{60}{7}+\left(\left(2\right)\sqrt{9}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{73}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{80}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{78}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{75}\right)+6-\left(\left(-\dfrac{79}{4}\right)\sqrt{12}\right)\right)-\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{15}{4}\right)\sqrt{75}-\dfrac{13}{3}+\left(-\dfrac{71}{9}\right)\sqrt{12}+\left(-\dfrac{77}{4}\right)\sqrt{75}+\left(\dfrac{8}{3}\right)\sqrt{27}\right)\\
&=&\left(\left(-20\right)\sqrt{3}+2+\dfrac{60}{7}+6-\left(\left(-\dfrac{37}{3}\right)\sqrt{3}\right)-\dfrac{219}{4}-\left(\left(\dfrac{240}{7}\right)\sqrt{3}\right)+\left(\left(-78\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)+6-\left(\left(-\dfrac{79}{2}\right)\sqrt{3}\right)\right)-\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\left(\dfrac{75}{4}\right)\sqrt{3}-\dfrac{13}{3}+\left(-\dfrac{142}{9}\right)\sqrt{3}+\left(-\dfrac{385}{4}\right)\sqrt{3}+\left(8\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{3463}{42}\right)\sqrt{3}-\dfrac{901}{28}\right)-\left(\left(-\dfrac{163}{2}\right)\sqrt{3}-\dfrac{13}{3}\right)\\
&=&\left(-\dfrac{3463}{42}\right)\sqrt{3}-\dfrac{901}{28}+\left(\dfrac{163}{2}\right)\sqrt{3}+\dfrac{13}{3}\\
&=&\left(-\dfrac{20}{21}\right)\sqrt{3}-\dfrac{2339}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-4\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{9}+\dfrac{60}{7}+\left(\left(2\right)\sqrt{9}\right)-\left(\left(-\dfrac{37}{6}\right)\sqrt{12}\right)-\left(\left(\dfrac{73}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{80}{7}\right)\sqrt{27}\right)+\left(\left(-\dfrac{78}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{5}\right)\sqrt{75}\right)+6-\left(\left(-\dfrac{79}{4}\right)\sqrt{12}\right)\right)\times\left(\left(\dfrac{17}{9}\right)\sqrt{12}+\left(\dfrac{15}{4}\right)\sqrt{75}-\dfrac{13}{3}+\left(-\dfrac{71}{9}\right)\sqrt{12}+\left(-\dfrac{77}{4}\right)\sqrt{75}+\left(\dfrac{8}{3}\right)\sqrt{27}\right)\\
&=&\left(\left(-20\right)\sqrt{3}+2+\dfrac{60}{7}+6-\left(\left(-\dfrac{37}{3}\right)\sqrt{3}\right)-\dfrac{219}{4}-\left(\left(\dfrac{240}{7}\right)\sqrt{3}\right)+\left(\left(-78\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)+6-\left(\left(-\dfrac{79}{2}\right)\sqrt{3}\right)\right)\times\left(\left(\dfrac{34}{9}\right)\sqrt{3}+\left(\dfrac{75}{4}\right)\sqrt{3}-\dfrac{13}{3}+\left(-\dfrac{142}{9}\right)\sqrt{3}+\left(-\dfrac{385}{4}\right)\sqrt{3}+\left(8\right)\sqrt{3}\right)\\
&=&\left(\left(-\dfrac{3463}{42}\right)\sqrt{3}-\dfrac{901}{28}\right)\left(\left(-\dfrac{163}{2}\right)\sqrt{3}-\dfrac{13}{3}\right)\\
&=&\left(\dfrac{564469}{84}\right)\sqrt{9}+\left(\dfrac{214549}{72}\right)\sqrt{3}+\dfrac{11713}{84}\\
&=&\dfrac{426280}{21}+\left(\dfrac{214549}{72}\right)\sqrt{3}\\
\end{eqnarray*}