L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{74}{9}\right)\sqrt{49}+\left(-9\right)\sqrt{175}+\left(-4\right)\sqrt{63}+\left(1\right)\sqrt{63}+\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{28}\right)\) et \( Y=-\dfrac{17}{7}+\left(\dfrac{70}{3}\right)\sqrt{28}+\left(0\right)\sqrt{49}+\left(\left(\dfrac{17}{6}\right)\sqrt{63}\right)-\left(\left(-7\right)\sqrt{175}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{49}+\left(-9\right)\sqrt{175}+\left(-4\right)\sqrt{63}+\left(1\right)\sqrt{63}+\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{28}\right)\right)+\left(-\dfrac{17}{7}+\left(\dfrac{70}{3}\right)\sqrt{28}+\left(0\right)\sqrt{49}+\left(\left(\dfrac{17}{6}\right)\sqrt{63}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\\
&=&\left(-\dfrac{518}{9}+\left(-45\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(3\right)\sqrt{7}-14-\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{7}\right)\right)+\left(-\dfrac{17}{7}+\left(\dfrac{140}{3}\right)\sqrt{7}+0+\left(\left(\dfrac{17}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\\
&=&-\dfrac{518}{9}+\left(-45\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(3\right)\sqrt{7}-14-\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{7}\right)-\dfrac{17}{7}+\left(\dfrac{140}{3}\right)\sqrt{7}+0+\left(\left(\dfrac{17}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\\
&=&-\dfrac{4661}{63}+\left(\dfrac{157}{4}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{49}+\left(-9\right)\sqrt{175}+\left(-4\right)\sqrt{63}+\left(1\right)\sqrt{63}+\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{28}\right)\right)-\left(-\dfrac{17}{7}+\left(\dfrac{70}{3}\right)\sqrt{28}+\left(0\right)\sqrt{49}+\left(\left(\dfrac{17}{6}\right)\sqrt{63}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\\
&=&\left(-\dfrac{518}{9}+\left(-45\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(3\right)\sqrt{7}-14-\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{7}\right)\right)-\left(-\dfrac{17}{7}+\left(\dfrac{140}{3}\right)\sqrt{7}+0+\left(\left(\dfrac{17}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{644}{9}+\left(-\dfrac{611}{12}\right)\sqrt{7}\right)-\left(-\dfrac{17}{7}+\left(\dfrac{541}{6}\right)\sqrt{7}\right)\\
&=&-\dfrac{644}{9}+\left(-\dfrac{611}{12}\right)\sqrt{7}+\dfrac{17}{7}+\left(-\dfrac{541}{6}\right)\sqrt{7}\\
&=&-\dfrac{4355}{63}+\left(-\dfrac{1693}{12}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{49}+\left(-9\right)\sqrt{175}+\left(-4\right)\sqrt{63}+\left(1\right)\sqrt{63}+\left(\left(-2\right)\sqrt{49}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{1}{8}\right)\sqrt{28}\right)\right)\times\left(-\dfrac{17}{7}+\left(\dfrac{70}{3}\right)\sqrt{28}+\left(0\right)\sqrt{49}+\left(\left(\dfrac{17}{6}\right)\sqrt{63}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\\
&=&\left(-\dfrac{518}{9}+\left(-45\right)\sqrt{7}+\left(-12\right)\sqrt{7}+\left(3\right)\sqrt{7}-14-\left(\left(-\dfrac{10}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{7}\right)\right)\times\left(-\dfrac{17}{7}+\left(\dfrac{140}{3}\right)\sqrt{7}+0+\left(\left(\dfrac{17}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{644}{9}+\left(-\dfrac{611}{12}\right)\sqrt{7}\right)\left(-\dfrac{17}{7}+\left(\dfrac{541}{6}\right)\sqrt{7}\right)\\
&=&\dfrac{1564}{9}+\left(-\dfrac{4784173}{756}\right)\sqrt{7}+\left(-\dfrac{330551}{72}\right)\sqrt{49}\\
&=&-\dfrac{255705}{8}+\left(-\dfrac{4784173}{756}\right)\sqrt{7}\\
\end{eqnarray*}