L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{29}{5}\right)\sqrt{27}-\dfrac{25}{3}+\left(-\dfrac{29}{4}\right)\sqrt{12}+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(\dfrac{40}{9}\right)\sqrt{9}+\left(\dfrac{45}{2}\right)\sqrt{9}+\left(-\dfrac{79}{4}\right)\sqrt{27}+\left(-\dfrac{34}{5}\right)\sqrt{75}+\left(\dfrac{56}{9}\right)\sqrt{12}+\dfrac{67}{5}-\left(\left(-\dfrac{71}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}\right)+\left(\dfrac{53}{3}\right)\sqrt{12}+4+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(-\dfrac{33}{4}\right)\sqrt{12}+\left(\dfrac{47}{5}\right)\sqrt{12}\) et \( Y=\left(4\right)\sqrt{27}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{29}{5}\right)\sqrt{27}-\dfrac{25}{3}+\left(-\dfrac{29}{4}\right)\sqrt{12}+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(\dfrac{40}{9}\right)\sqrt{9}+\left(\dfrac{45}{2}\right)\sqrt{9}+\left(-\dfrac{79}{4}\right)\sqrt{27}+\left(-\dfrac{34}{5}\right)\sqrt{75}+\left(\dfrac{56}{9}\right)\sqrt{12}+\dfrac{67}{5}-\left(\left(-\dfrac{71}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}\right)+\left(\dfrac{53}{3}\right)\sqrt{12}+4+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(-\dfrac{33}{4}\right)\sqrt{12}+\left(\dfrac{47}{5}\right)\sqrt{12}\right)+\left(\left(4\right)\sqrt{27}\right)\\
&=&\left(\left(\dfrac{87}{5}\right)\sqrt{3}-\dfrac{25}{3}+\left(-\dfrac{29}{2}\right)\sqrt{3}+\left(\dfrac{204}{7}\right)\sqrt{3}+\dfrac{40}{3}+\dfrac{135}{2}+\left(-\dfrac{237}{4}\right)\sqrt{3}+\left(-34\right)\sqrt{3}+\left(\dfrac{112}{9}\right)\sqrt{3}+\dfrac{67}{5}+\dfrac{213}{4}-\left(\left(9\right)\sqrt{3}\right)+\left(\dfrac{106}{3}\right)\sqrt{3}+4+\left(\dfrac{204}{7}\right)\sqrt{3}+\left(-\dfrac{33}{2}\right)\sqrt{3}+\left(\dfrac{94}{5}\right)\sqrt{3}\right)+\left(\left(12\right)\sqrt{3}\right)\\
&=&\left(\dfrac{87}{5}\right)\sqrt{3}-\dfrac{25}{3}+\left(-\dfrac{29}{2}\right)\sqrt{3}+\left(\dfrac{204}{7}\right)\sqrt{3}+\dfrac{40}{3}+\dfrac{135}{2}+\left(-\dfrac{237}{4}\right)\sqrt{3}+\left(-34\right)\sqrt{3}+\left(\dfrac{112}{9}\right)\sqrt{3}+\dfrac{67}{5}+\dfrac{213}{4}-\left(\left(9\right)\sqrt{3}\right)+\left(\dfrac{106}{3}\right)\sqrt{3}+4+\left(\dfrac{204}{7}\right)\sqrt{3}+\left(-\dfrac{33}{2}\right)\sqrt{3}+\left(\dfrac{94}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}\\
&=&\left(\dfrac{26477}{1260}\right)\sqrt{3}+\dfrac{2863}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{29}{5}\right)\sqrt{27}-\dfrac{25}{3}+\left(-\dfrac{29}{4}\right)\sqrt{12}+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(\dfrac{40}{9}\right)\sqrt{9}+\left(\dfrac{45}{2}\right)\sqrt{9}+\left(-\dfrac{79}{4}\right)\sqrt{27}+\left(-\dfrac{34}{5}\right)\sqrt{75}+\left(\dfrac{56}{9}\right)\sqrt{12}+\dfrac{67}{5}-\left(\left(-\dfrac{71}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}\right)+\left(\dfrac{53}{3}\right)\sqrt{12}+4+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(-\dfrac{33}{4}\right)\sqrt{12}+\left(\dfrac{47}{5}\right)\sqrt{12}\right)-\left(\left(4\right)\sqrt{27}\right)\\
&=&\left(\left(\dfrac{87}{5}\right)\sqrt{3}-\dfrac{25}{3}+\left(-\dfrac{29}{2}\right)\sqrt{3}+\left(\dfrac{204}{7}\right)\sqrt{3}+\dfrac{40}{3}+\dfrac{135}{2}+\left(-\dfrac{237}{4}\right)\sqrt{3}+\left(-34\right)\sqrt{3}+\left(\dfrac{112}{9}\right)\sqrt{3}+\dfrac{67}{5}+\dfrac{213}{4}-\left(\left(9\right)\sqrt{3}\right)+\left(\dfrac{106}{3}\right)\sqrt{3}+4+\left(\dfrac{204}{7}\right)\sqrt{3}+\left(-\dfrac{33}{2}\right)\sqrt{3}+\left(\dfrac{94}{5}\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{11357}{1260}\right)\sqrt{3}+\dfrac{2863}{20}\right)-\left(\left(12\right)\sqrt{3}\right)\\
&=&\left(\dfrac{11357}{1260}\right)\sqrt{3}+\dfrac{2863}{20}+\left(-12\right)\sqrt{3}\\
&=&\left(-\dfrac{3763}{1260}\right)\sqrt{3}+\dfrac{2863}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{29}{5}\right)\sqrt{27}-\dfrac{25}{3}+\left(-\dfrac{29}{4}\right)\sqrt{12}+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(\dfrac{40}{9}\right)\sqrt{9}+\left(\dfrac{45}{2}\right)\sqrt{9}+\left(-\dfrac{79}{4}\right)\sqrt{27}+\left(-\dfrac{34}{5}\right)\sqrt{75}+\left(\dfrac{56}{9}\right)\sqrt{12}+\dfrac{67}{5}-\left(\left(-\dfrac{71}{4}\right)\sqrt{9}\right)-\left(\left(3\right)\sqrt{27}\right)+\left(\dfrac{53}{3}\right)\sqrt{12}+4+\left(\dfrac{68}{7}\right)\sqrt{27}+\left(-\dfrac{33}{4}\right)\sqrt{12}+\left(\dfrac{47}{5}\right)\sqrt{12}\right)\times\left(\left(4\right)\sqrt{27}\right)\\
&=&\left(\left(\dfrac{87}{5}\right)\sqrt{3}-\dfrac{25}{3}+\left(-\dfrac{29}{2}\right)\sqrt{3}+\left(\dfrac{204}{7}\right)\sqrt{3}+\dfrac{40}{3}+\dfrac{135}{2}+\left(-\dfrac{237}{4}\right)\sqrt{3}+\left(-34\right)\sqrt{3}+\left(\dfrac{112}{9}\right)\sqrt{3}+\dfrac{67}{5}+\dfrac{213}{4}-\left(\left(9\right)\sqrt{3}\right)+\left(\dfrac{106}{3}\right)\sqrt{3}+4+\left(\dfrac{204}{7}\right)\sqrt{3}+\left(-\dfrac{33}{2}\right)\sqrt{3}+\left(\dfrac{94}{5}\right)\sqrt{3}\right)\times\left(\left(12\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{11357}{1260}\right)\sqrt{3}+\dfrac{2863}{20}\right)\left(\left(12\right)\sqrt{3}\right)\\
&=&\left(\dfrac{11357}{105}\right)\sqrt{9}+\left(\dfrac{8589}{5}\right)\sqrt{3}\\
&=&\dfrac{11357}{35}+\left(\dfrac{8589}{5}\right)\sqrt{3}\\
\end{eqnarray*}