L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-2\right)\sqrt{125}\right)-\left(\left(-5\right)\sqrt{25}+\left(-5\right)\sqrt{25}+\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{5}{6}\right)\sqrt{25}-6+\left(-\dfrac{69}{7}\right)\sqrt{20}+\dfrac{23}{3}\right)-\left(\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{125}\right)\right)\) et \( Y=\left(\dfrac{41}{9}\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-2\right)\sqrt{125}\right)-\left(\left(-5\right)\sqrt{25}+\left(-5\right)\sqrt{25}+\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{5}{6}\right)\sqrt{25}-6+\left(-\dfrac{69}{7}\right)\sqrt{20}+\dfrac{23}{3}\right)-\left(\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{125}\right)\right)\right)+\left(\left(\dfrac{41}{9}\right)\sqrt{25}\right)\\
&=&\left(\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-10\right)\sqrt{5}\right)-\left(-25-25+\left(-20\right)\sqrt{5}\right)-\left(-\dfrac{25}{6}-6+\left(-\dfrac{138}{7}\right)\sqrt{5}+\dfrac{23}{3}\right)-\left(\dfrac{35}{3}-\left(\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)\right)\right)+\left(\dfrac{205}{9}\right)\\
&=&\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-10\right)\sqrt{5}\right)-\left(-25-25+\left(-20\right)\sqrt{5}\right)-\left(-\dfrac{25}{6}-6+\left(-\dfrac{138}{7}\right)\sqrt{5}+\dfrac{23}{3}\right)-\left(\dfrac{35}{3}-\left(\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)\right)+\dfrac{205}{9}\\
&=&\dfrac{5429}{72}+\left(\dfrac{185}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-2\right)\sqrt{125}\right)-\left(\left(-5\right)\sqrt{25}+\left(-5\right)\sqrt{25}+\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{5}{6}\right)\sqrt{25}-6+\left(-\dfrac{69}{7}\right)\sqrt{20}+\dfrac{23}{3}\right)-\left(\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{125}\right)\right)\right)-\left(\left(\dfrac{41}{9}\right)\sqrt{25}\right)\\
&=&\left(\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-10\right)\sqrt{5}\right)-\left(-25-25+\left(-20\right)\sqrt{5}\right)-\left(-\dfrac{25}{6}-6+\left(-\dfrac{138}{7}\right)\sqrt{5}+\dfrac{23}{3}\right)-\left(\dfrac{35}{3}-\left(\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)\right)\right)-\left(\dfrac{205}{9}\right)\\
&=&\left(\dfrac{421}{8}+\left(\dfrac{185}{14}\right)\sqrt{5}\right)-\left(\dfrac{205}{9}\right)\\
&=&\dfrac{421}{8}+\left(\dfrac{185}{14}\right)\sqrt{5}+-\dfrac{205}{9}\\
&=&\dfrac{2149}{72}+\left(\dfrac{185}{14}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-2\right)\sqrt{125}\right)-\left(\left(-5\right)\sqrt{25}+\left(-5\right)\sqrt{25}+\left(-\dfrac{20}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{5}{6}\right)\sqrt{25}-6+\left(-\dfrac{69}{7}\right)\sqrt{20}+\dfrac{23}{3}\right)-\left(\left(\left(\dfrac{7}{3}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{45}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{125}\right)\right)\right)\times\left(\left(\dfrac{41}{9}\right)\sqrt{25}\right)\\
&=&\left(\left(\dfrac{33}{8}+\dfrac{23}{3}+\left(-10\right)\sqrt{5}\right)-\left(-25-25+\left(-20\right)\sqrt{5}\right)-\left(-\dfrac{25}{6}-6+\left(-\dfrac{138}{7}\right)\sqrt{5}+\dfrac{23}{3}\right)-\left(\dfrac{35}{3}-\left(\left(-9\right)\sqrt{5}\right)-\left(\left(-\dfrac{15}{2}\right)\sqrt{5}\right)\right)\right)\times\left(\dfrac{205}{9}\right)\\
&=&\left(\dfrac{421}{8}+\left(\dfrac{185}{14}\right)\sqrt{5}\right)\left(\dfrac{205}{9}\right)\\
&=&\dfrac{86305}{72}+\left(\dfrac{37925}{126}\right)\sqrt{5}\\
&=&\dfrac{86305}{72}+\left(\dfrac{37925}{126}\right)\sqrt{5}\\
\end{eqnarray*}