L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{43}{2}\right)\sqrt{27}+\left(-\dfrac{22}{5}\right)\sqrt{27}+\left(6\right)\sqrt{12}+\left(9\right)\sqrt{9}+\left(\dfrac{31}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{36}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{25}{6}\right)\sqrt{27}\right)\right)\) et \( Y=\left(\left(\dfrac{47}{7}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{71}{3}\right)\sqrt{75}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{8}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{12}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{27}+\left(-\dfrac{22}{5}\right)\sqrt{27}+\left(6\right)\sqrt{12}+\left(9\right)\sqrt{9}+\left(\dfrac{31}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{36}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{25}{6}\right)\sqrt{27}\right)\right)\right)+\left(\left(\left(\dfrac{47}{7}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{71}{3}\right)\sqrt{75}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{8}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(-\dfrac{129}{2}\right)\sqrt{3}+\left(-\dfrac{66}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+27+\dfrac{93}{2}\right)-\left(\left(\left(-36\right)\sqrt{3}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{3}\right)\right)\right)+\left(\left(\left(\dfrac{235}{7}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{355}{3}\right)\sqrt{3}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{24}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{153}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{129}{2}\right)\sqrt{3}+\left(-\dfrac{66}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+27+\dfrac{93}{2}\right)-\left(\left(\left(-36\right)\sqrt{3}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{3}\right)\right)+\left(\left(\dfrac{235}{7}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{355}{3}\right)\sqrt{3}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{24}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{153}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)\\
&=&\left(\dfrac{39061}{210}\right)\sqrt{3}+\dfrac{873}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{27}+\left(-\dfrac{22}{5}\right)\sqrt{27}+\left(6\right)\sqrt{12}+\left(9\right)\sqrt{9}+\left(\dfrac{31}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{36}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{25}{6}\right)\sqrt{27}\right)\right)\right)-\left(\left(\left(\dfrac{47}{7}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{71}{3}\right)\sqrt{75}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{8}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(-\dfrac{129}{2}\right)\sqrt{3}+\left(-\dfrac{66}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+27+\dfrac{93}{2}\right)-\left(\left(\left(-36\right)\sqrt{3}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{3}\right)\right)\right)-\left(\left(\left(\dfrac{235}{7}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{355}{3}\right)\sqrt{3}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{24}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{153}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{86}{5}\right)\sqrt{3}+\dfrac{147}{2}\right)-\left(\left(\dfrac{42673}{210}\right)\sqrt{3}+\dfrac{69}{5}\right)\\
&=&\left(-\dfrac{86}{5}\right)\sqrt{3}+\dfrac{147}{2}+\left(-\dfrac{42673}{210}\right)\sqrt{3}-\dfrac{69}{5}\\
&=&\left(-\dfrac{9257}{42}\right)\sqrt{3}+\dfrac{597}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{43}{2}\right)\sqrt{27}+\left(-\dfrac{22}{5}\right)\sqrt{27}+\left(6\right)\sqrt{12}+\left(9\right)\sqrt{9}+\left(\dfrac{31}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{36}{5}\right)\sqrt{75}\right)-\left(\left(\dfrac{25}{6}\right)\sqrt{27}\right)\right)\right)\times\left(\left(\left(\dfrac{47}{7}\right)\sqrt{75}\right)-\left(\left(8\right)\sqrt{27}\right)-\left(\left(-\dfrac{71}{3}\right)\sqrt{75}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{8}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{12}\right)\right)\\
&=&\left(\left(\left(-\dfrac{129}{2}\right)\sqrt{3}+\left(-\dfrac{66}{5}\right)\sqrt{3}+\left(12\right)\sqrt{3}+27+\dfrac{93}{2}\right)-\left(\left(\left(-36\right)\sqrt{3}\right)-\left(\left(\dfrac{25}{2}\right)\sqrt{3}\right)\right)\right)\times\left(\left(\left(\dfrac{235}{7}\right)\sqrt{3}\right)-\left(\left(24\right)\sqrt{3}\right)-\left(\left(-\dfrac{355}{3}\right)\sqrt{3}\right)+\dfrac{79}{5}-2-\left(\left(-\dfrac{24}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{153}{2}\right)\sqrt{3}\right)-\left(\left(6\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{86}{5}\right)\sqrt{3}+\dfrac{147}{2}\right)\left(\left(\dfrac{42673}{210}\right)\sqrt{3}+\dfrac{69}{5}\right)\\
&=&\left(-\dfrac{1834939}{525}\right)\sqrt{9}+\left(\dfrac{1469819}{100}\right)\sqrt{3}+\dfrac{10143}{10}\\
&=&-\dfrac{3314873}{350}+\left(\dfrac{1469819}{100}\right)\sqrt{3}\\
\end{eqnarray*}