L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=8\) et \( Y=\dfrac{5}{8}+\left(0\right)\sqrt{4}+\left(-\dfrac{45}{2}\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(2\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{8}+\left(-\dfrac{64}{3}\right)\sqrt{8}+\left(\dfrac{46}{7}\right)\sqrt{18}+\left(-\dfrac{27}{2}\right)\sqrt{50}+\left(\left(-\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(8\right)+\left(\dfrac{5}{8}+\left(0\right)\sqrt{4}+\left(-\dfrac{45}{2}\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(2\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{8}+\left(-\dfrac{64}{3}\right)\sqrt{8}+\left(\dfrac{46}{7}\right)\sqrt{18}+\left(-\dfrac{27}{2}\right)\sqrt{50}+\left(\left(-\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)\right)\\
&=&\left(8\right)+\left(\dfrac{5}{8}+0+\left(-45\right)\sqrt{2}+\left(-5\right)\sqrt{2}+4+\left(27\right)\sqrt{2}+\left(-\dfrac{128}{3}\right)\sqrt{2}+\left(\dfrac{138}{7}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\left(-\dfrac{135}{2}\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)\right)\\
&=&8+\dfrac{5}{8}+0+\left(-45\right)\sqrt{2}+\left(-5\right)\sqrt{2}+4+\left(27\right)\sqrt{2}+\left(-\dfrac{128}{3}\right)\sqrt{2}+\left(\dfrac{138}{7}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\left(-\dfrac{135}{2}\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)\\
&=&\dfrac{101}{8}+\left(-\dfrac{4367}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(8\right)-\left(\dfrac{5}{8}+\left(0\right)\sqrt{4}+\left(-\dfrac{45}{2}\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(2\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{8}+\left(-\dfrac{64}{3}\right)\sqrt{8}+\left(\dfrac{46}{7}\right)\sqrt{18}+\left(-\dfrac{27}{2}\right)\sqrt{50}+\left(\left(-\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)\right)\\
&=&\left(8\right)-\left(\dfrac{5}{8}+0+\left(-45\right)\sqrt{2}+\left(-5\right)\sqrt{2}+4+\left(27\right)\sqrt{2}+\left(-\dfrac{128}{3}\right)\sqrt{2}+\left(\dfrac{138}{7}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\left(-\dfrac{135}{2}\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)\right)\\
&=&\left(8\right)-\left(\dfrac{37}{8}+\left(-\dfrac{4367}{21}\right)\sqrt{2}\right)\\
&=&8+-\dfrac{37}{8}+\left(\dfrac{4367}{21}\right)\sqrt{2}\\
&=&\dfrac{27}{8}+\left(\dfrac{4367}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(8\right)\times\left(\dfrac{5}{8}+\left(0\right)\sqrt{4}+\left(-\dfrac{45}{2}\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(2\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{8}+\left(-\dfrac{64}{3}\right)\sqrt{8}+\left(\dfrac{46}{7}\right)\sqrt{18}+\left(-\dfrac{27}{2}\right)\sqrt{50}+\left(\left(-\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)\right)\\
&=&\left(8\right)\times\left(\dfrac{5}{8}+0+\left(-45\right)\sqrt{2}+\left(-5\right)\sqrt{2}+4+\left(27\right)\sqrt{2}+\left(-\dfrac{128}{3}\right)\sqrt{2}+\left(\dfrac{138}{7}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\left(-\dfrac{135}{2}\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)\right)\\
&=&\left(8\right)\left(\dfrac{37}{8}+\left(-\dfrac{4367}{21}\right)\sqrt{2}\right)\\
&=&37+\left(-\dfrac{34936}{21}\right)\sqrt{2}\\
&=&37+\left(-\dfrac{34936}{21}\right)\sqrt{2}\\
\end{eqnarray*}