L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-6\right)\sqrt{175}\right)-\left(\left(8\right)\sqrt{175}+\left(\dfrac{3}{4}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)\) et \( Y=\left(\left(\left(6\right)\sqrt{28}\right)-\dfrac{16}{3}-\left(\left(\dfrac{37}{2}\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{63}\right)-\left(-1+\left(-7\right)\sqrt{63}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(8\right)\sqrt{175}+\left(\dfrac{3}{4}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)\right)+\left(\left(\left(\left(6\right)\sqrt{28}\right)-\dfrac{16}{3}-\left(\left(\dfrac{37}{2}\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{63}\right)-\left(-1+\left(-7\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\left(-30\right)\sqrt{7}\right)-\left(\left(40\right)\sqrt{7}+\left(\dfrac{15}{4}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)\right)+\left(\left(\left(\left(12\right)\sqrt{7}\right)-\dfrac{16}{3}-\dfrac{259}{2}\right)-\left(\left(-\dfrac{105}{2}\right)\sqrt{7}\right)-\left(-1+\left(-21\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-30\right)\sqrt{7}\right)-\left(\left(40\right)\sqrt{7}+\left(\dfrac{15}{4}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)+\left(\left(\left(12\right)\sqrt{7}\right)-\dfrac{16}{3}-\dfrac{259}{2}\right)-\left(\left(-\dfrac{105}{2}\right)\sqrt{7}\right)-\left(-1+\left(-21\right)\sqrt{7}\right)\\
&=&\left(\dfrac{71}{4}\right)\sqrt{7}-\dfrac{803}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(8\right)\sqrt{175}+\left(\dfrac{3}{4}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)\right)-\left(\left(\left(\left(6\right)\sqrt{28}\right)-\dfrac{16}{3}-\left(\left(\dfrac{37}{2}\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{63}\right)-\left(-1+\left(-7\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\left(-30\right)\sqrt{7}\right)-\left(\left(40\right)\sqrt{7}+\left(\dfrac{15}{4}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)\right)-\left(\left(\left(\left(12\right)\sqrt{7}\right)-\dfrac{16}{3}-\dfrac{259}{2}\right)-\left(\left(-\dfrac{105}{2}\right)\sqrt{7}\right)-\left(-1+\left(-21\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{271}{4}\right)\sqrt{7}\right)-\left(\left(\dfrac{171}{2}\right)\sqrt{7}-\dfrac{803}{6}\right)\\
&=&\left(-\dfrac{271}{4}\right)\sqrt{7}+\left(-\dfrac{171}{2}\right)\sqrt{7}+\dfrac{803}{6}\\
&=&\left(-\dfrac{613}{4}\right)\sqrt{7}+\dfrac{803}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-6\right)\sqrt{175}\right)-\left(\left(8\right)\sqrt{175}+\left(\dfrac{3}{4}\right)\sqrt{175}\right)-\left(\left(-2\right)\sqrt{63}\right)\right)\times\left(\left(\left(\left(6\right)\sqrt{28}\right)-\dfrac{16}{3}-\left(\left(\dfrac{37}{2}\right)\sqrt{49}\right)\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{63}\right)-\left(-1+\left(-7\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\left(-30\right)\sqrt{7}\right)-\left(\left(40\right)\sqrt{7}+\left(\dfrac{15}{4}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)\right)\times\left(\left(\left(\left(12\right)\sqrt{7}\right)-\dfrac{16}{3}-\dfrac{259}{2}\right)-\left(\left(-\dfrac{105}{2}\right)\sqrt{7}\right)-\left(-1+\left(-21\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{271}{4}\right)\sqrt{7}\right)\left(\left(\dfrac{171}{2}\right)\sqrt{7}-\dfrac{803}{6}\right)\\
&=&\left(-\dfrac{46341}{8}\right)\sqrt{49}+\left(\dfrac{217613}{24}\right)\sqrt{7}\\
&=&-\dfrac{324387}{8}+\left(\dfrac{217613}{24}\right)\sqrt{7}\\
\end{eqnarray*}