L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-7+\left(8\right)\sqrt{8}+\left(\dfrac{71}{6}\right)\sqrt{4}+\left(\dfrac{70}{9}\right)\sqrt{4}+\left(0\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\dfrac{15}{7}\right)\sqrt{4}\) et \( Y=\left(-\dfrac{62}{9}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-7+\left(8\right)\sqrt{8}+\left(\dfrac{71}{6}\right)\sqrt{4}+\left(\dfrac{70}{9}\right)\sqrt{4}+\left(0\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\dfrac{15}{7}\right)\sqrt{4}\right)+\left(\left(-\dfrac{62}{9}\right)\sqrt{50}\right)\\
&=&\left(-7+\left(16\right)\sqrt{2}+\dfrac{71}{3}+\dfrac{140}{9}+\left(0\right)\sqrt{2}+16+\dfrac{30}{7}\right)+\left(\left(-\dfrac{310}{9}\right)\sqrt{2}\right)\\
&=&-7+\left(16\right)\sqrt{2}+\dfrac{71}{3}+\dfrac{140}{9}+\left(0\right)\sqrt{2}+16+\dfrac{30}{7}+\left(-\dfrac{310}{9}\right)\sqrt{2}\\
&=&\dfrac{3308}{63}+\left(-\dfrac{166}{9}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-7+\left(8\right)\sqrt{8}+\left(\dfrac{71}{6}\right)\sqrt{4}+\left(\dfrac{70}{9}\right)\sqrt{4}+\left(0\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\dfrac{15}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{62}{9}\right)\sqrt{50}\right)\\
&=&\left(-7+\left(16\right)\sqrt{2}+\dfrac{71}{3}+\dfrac{140}{9}+\left(0\right)\sqrt{2}+16+\dfrac{30}{7}\right)-\left(\left(-\dfrac{310}{9}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{3308}{63}+\left(16\right)\sqrt{2}\right)-\left(\left(-\dfrac{310}{9}\right)\sqrt{2}\right)\\
&=&\dfrac{3308}{63}+\left(16\right)\sqrt{2}+\left(\dfrac{310}{9}\right)\sqrt{2}\\
&=&\dfrac{3308}{63}+\left(\dfrac{454}{9}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-7+\left(8\right)\sqrt{8}+\left(\dfrac{71}{6}\right)\sqrt{4}+\left(\dfrac{70}{9}\right)\sqrt{4}+\left(0\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\dfrac{15}{7}\right)\sqrt{4}\right)\times\left(\left(-\dfrac{62}{9}\right)\sqrt{50}\right)\\
&=&\left(-7+\left(16\right)\sqrt{2}+\dfrac{71}{3}+\dfrac{140}{9}+\left(0\right)\sqrt{2}+16+\dfrac{30}{7}\right)\times\left(\left(-\dfrac{310}{9}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{3308}{63}+\left(16\right)\sqrt{2}\right)\left(\left(-\dfrac{310}{9}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{1025480}{567}\right)\sqrt{2}+\left(-\dfrac{4960}{9}\right)\sqrt{4}\\
&=&\left(-\dfrac{1025480}{567}\right)\sqrt{2}-\dfrac{9920}{9}\\
\end{eqnarray*}