L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{50}{7}\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{12}+\left(-1\right)\sqrt{12}+\left(\dfrac{22}{9}\right)\sqrt{9}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{28}{3}\right)\sqrt{75}+\left(\dfrac{68}{7}\right)\sqrt{75}+\left(\left(\dfrac{22}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{75}\right)\) et \( Y=\left(\dfrac{56}{3}\right)\sqrt{27}\) . 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{50}{7}\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{12}+\left(-1\right)\sqrt{12}+\left(\dfrac{22}{9}\right)\sqrt{9}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{28}{3}\right)\sqrt{75}+\left(\dfrac{68}{7}\right)\sqrt{75}+\left(\left(\dfrac{22}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{75}\right)\right)+\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)\\
&=&\left(-\dfrac{150}{7}+\left(\dfrac{55}{2}\right)\sqrt{3}+\left(\dfrac{4}{3}\right)\sqrt{3}+\left(-2\right)\sqrt{3}+\dfrac{22}{3}-\dfrac{153}{5}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{340}{7}\right)\sqrt{3}+\dfrac{22}{3}-\left(\left(\dfrac{47}{2}\right)\sqrt{3}\right)-3-\left(\left(-\dfrac{140}{3}\right)\sqrt{3}\right)\right)+\left(\left(56\right)\sqrt{3}\right)\\
&=&-\dfrac{150}{7}+\left(\dfrac{55}{2}\right)\sqrt{3}+\left(\dfrac{4}{3}\right)\sqrt{3}+\left(-2\right)\sqrt{3}+\dfrac{22}{3}-\dfrac{153}{5}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{340}{7}\right)\sqrt{3}+\dfrac{22}{3}-\left(\left(\dfrac{47}{2}\right)\sqrt{3}\right)-3-\left(\left(-\dfrac{140}{3}\right)\sqrt{3}\right)+\left(56\right)\sqrt{3}\\
&=&-\dfrac{4238}{105}+\left(\dfrac{2546}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{50}{7}\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{12}+\left(-1\right)\sqrt{12}+\left(\dfrac{22}{9}\right)\sqrt{9}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{28}{3}\right)\sqrt{75}+\left(\dfrac{68}{7}\right)\sqrt{75}+\left(\left(\dfrac{22}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{75}\right)\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)\\
&=&\left(-\dfrac{150}{7}+\left(\dfrac{55}{2}\right)\sqrt{3}+\left(\dfrac{4}{3}\right)\sqrt{3}+\left(-2\right)\sqrt{3}+\dfrac{22}{3}-\dfrac{153}{5}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{340}{7}\right)\sqrt{3}+\dfrac{22}{3}-\left(\left(\dfrac{47}{2}\right)\sqrt{3}\right)-3-\left(\left(-\dfrac{140}{3}\right)\sqrt{3}\right)\right)-\left(\left(56\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{4238}{105}+\left(\dfrac{1370}{21}\right)\sqrt{3}\right)-\left(\left(56\right)\sqrt{3}\right)\\
&=&-\dfrac{4238}{105}+\left(\dfrac{1370}{21}\right)\sqrt{3}+\left(-56\right)\sqrt{3}\\
&=&-\dfrac{4238}{105}+\left(\dfrac{194}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{50}{7}\right)\sqrt{9}+\left(\dfrac{11}{2}\right)\sqrt{75}+\left(\dfrac{2}{3}\right)\sqrt{12}+\left(-1\right)\sqrt{12}+\left(\dfrac{22}{9}\right)\sqrt{9}+\left(-\dfrac{51}{5}\right)\sqrt{9}+\left(\dfrac{8}{3}\right)\sqrt{75}+\left(-\dfrac{28}{3}\right)\sqrt{75}+\left(\dfrac{68}{7}\right)\sqrt{75}+\left(\left(\dfrac{22}{9}\right)\sqrt{9}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{27}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{75}\right)\right)\times\left(\left(\dfrac{56}{3}\right)\sqrt{27}\right)\\
&=&\left(-\dfrac{150}{7}+\left(\dfrac{55}{2}\right)\sqrt{3}+\left(\dfrac{4}{3}\right)\sqrt{3}+\left(-2\right)\sqrt{3}+\dfrac{22}{3}-\dfrac{153}{5}+\left(\dfrac{40}{3}\right)\sqrt{3}+\left(-\dfrac{140}{3}\right)\sqrt{3}+\left(\dfrac{340}{7}\right)\sqrt{3}+\dfrac{22}{3}-\left(\left(\dfrac{47}{2}\right)\sqrt{3}\right)-3-\left(\left(-\dfrac{140}{3}\right)\sqrt{3}\right)\right)\times\left(\left(56\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{4238}{105}+\left(\dfrac{1370}{21}\right)\sqrt{3}\right)\left(\left(56\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{33904}{15}\right)\sqrt{3}+\left(\dfrac{10960}{3}\right)\sqrt{9}\\
&=&\left(-\dfrac{33904}{15}\right)\sqrt{3}+10960\\
\end{eqnarray*}