L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{25}{9}\right)\sqrt{49}\) et \( Y=-\dfrac{29}{3}-5-8+\left(\left(-2\right)\sqrt{28}\right)-4-\left(\left(\dfrac{29}{4}\right)\sqrt{28}\right)+\left(0\right)\sqrt{28}+\left(-\dfrac{5}{2}\right)\sqrt{63}+\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{35}{4}\right)\sqrt{175}+\left(-\dfrac{69}{2}\right)\sqrt{63}\) . 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{25}{9}\right)\sqrt{49}\right)+\left(-\dfrac{29}{3}-5-8+\left(\left(-2\right)\sqrt{28}\right)-4-\left(\left(\dfrac{29}{4}\right)\sqrt{28}\right)+\left(0\right)\sqrt{28}+\left(-\dfrac{5}{2}\right)\sqrt{63}+\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{35}{4}\right)\sqrt{175}+\left(-\dfrac{69}{2}\right)\sqrt{63}\right)\\
&=&\left(\dfrac{175}{9}\right)+\left(-\dfrac{29}{3}-5-8+\left(\left(-4\right)\sqrt{7}\right)-4-\left(\left(\dfrac{29}{2}\right)\sqrt{7}\right)+\left(0\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}+\left(-10\right)\sqrt{7}+\left(\dfrac{175}{4}\right)\sqrt{7}+\left(-\dfrac{207}{2}\right)\sqrt{7}\right)\\
&=&\dfrac{175}{9}-\dfrac{29}{3}-5-8+\left(\left(-4\right)\sqrt{7}\right)-4-\left(\left(\dfrac{29}{2}\right)\sqrt{7}\right)+\left(0\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}+\left(-10\right)\sqrt{7}+\left(\dfrac{175}{4}\right)\sqrt{7}+\left(-\dfrac{207}{2}\right)\sqrt{7}\\
&=&-\dfrac{65}{9}+\left(-\dfrac{383}{4}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{25}{9}\right)\sqrt{49}\right)-\left(-\dfrac{29}{3}-5-8+\left(\left(-2\right)\sqrt{28}\right)-4-\left(\left(\dfrac{29}{4}\right)\sqrt{28}\right)+\left(0\right)\sqrt{28}+\left(-\dfrac{5}{2}\right)\sqrt{63}+\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{35}{4}\right)\sqrt{175}+\left(-\dfrac{69}{2}\right)\sqrt{63}\right)\\
&=&\left(\dfrac{175}{9}\right)-\left(-\dfrac{29}{3}-5-8+\left(\left(-4\right)\sqrt{7}\right)-4-\left(\left(\dfrac{29}{2}\right)\sqrt{7}\right)+\left(0\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}+\left(-10\right)\sqrt{7}+\left(\dfrac{175}{4}\right)\sqrt{7}+\left(-\dfrac{207}{2}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{175}{9}\right)-\left(-\dfrac{80}{3}+\left(-\dfrac{383}{4}\right)\sqrt{7}\right)\\
&=&\dfrac{175}{9}+\dfrac{80}{3}+\left(\dfrac{383}{4}\right)\sqrt{7}\\
&=&\dfrac{415}{9}+\left(\dfrac{383}{4}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{25}{9}\right)\sqrt{49}\right)\times\left(-\dfrac{29}{3}-5-8+\left(\left(-2\right)\sqrt{28}\right)-4-\left(\left(\dfrac{29}{4}\right)\sqrt{28}\right)+\left(0\right)\sqrt{28}+\left(-\dfrac{5}{2}\right)\sqrt{63}+\left(-\dfrac{10}{3}\right)\sqrt{63}+\left(\dfrac{35}{4}\right)\sqrt{175}+\left(-\dfrac{69}{2}\right)\sqrt{63}\right)\\
&=&\left(\dfrac{175}{9}\right)\times\left(-\dfrac{29}{3}-5-8+\left(\left(-4\right)\sqrt{7}\right)-4-\left(\left(\dfrac{29}{2}\right)\sqrt{7}\right)+\left(0\right)\sqrt{7}+\left(-\dfrac{15}{2}\right)\sqrt{7}+\left(-10\right)\sqrt{7}+\left(\dfrac{175}{4}\right)\sqrt{7}+\left(-\dfrac{207}{2}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{175}{9}\right)\left(-\dfrac{80}{3}+\left(-\dfrac{383}{4}\right)\sqrt{7}\right)\\
&=&-\dfrac{14000}{27}+\left(-\dfrac{67025}{36}\right)\sqrt{7}\\
&=&-\dfrac{14000}{27}+\left(-\dfrac{67025}{36}\right)\sqrt{7}\\
\end{eqnarray*}