L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{175}\right)-\left(\left(-3\right)\sqrt{28}\right)+\left(-\dfrac{33}{5}\right)\sqrt{49}-\dfrac{43}{2}+\left(-\dfrac{79}{2}\right)\sqrt{28}+\left(-\dfrac{1}{3}\right)\sqrt{63}\) et \( Y=\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{25}{2}\right)\sqrt{63}+\left(-4\right)\sqrt{28}-\dfrac{22}{3}+\left(\dfrac{51}{2}\right)\sqrt{175}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{175}\right)-\left(\left(-3\right)\sqrt{28}\right)+\left(-\dfrac{33}{5}\right)\sqrt{49}-\dfrac{43}{2}+\left(-\dfrac{79}{2}\right)\sqrt{28}+\left(-\dfrac{1}{3}\right)\sqrt{63}\right)+\left(\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{25}{2}\right)\sqrt{63}+\left(-4\right)\sqrt{28}-\dfrac{22}{3}+\left(\dfrac{51}{2}\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{275}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{231}{5}-\dfrac{43}{2}+\left(-79\right)\sqrt{7}+\left(-1\right)\sqrt{7}\right)+\left(\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{75}{2}\right)\sqrt{7}+\left(-8\right)\sqrt{7}-\dfrac{22}{3}+\left(\dfrac{255}{2}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{275}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{231}{5}-\dfrac{43}{2}+\left(-79\right)\sqrt{7}+\left(-1\right)\sqrt{7}+\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{75}{2}\right)\sqrt{7}+\left(-8\right)\sqrt{7}-\dfrac{22}{3}+\left(\dfrac{255}{2}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{671}{3}\right)\sqrt{7}-\dfrac{411}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{175}\right)-\left(\left(-3\right)\sqrt{28}\right)+\left(-\dfrac{33}{5}\right)\sqrt{49}-\dfrac{43}{2}+\left(-\dfrac{79}{2}\right)\sqrt{28}+\left(-\dfrac{1}{3}\right)\sqrt{63}\right)-\left(\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{25}{2}\right)\sqrt{63}+\left(-4\right)\sqrt{28}-\dfrac{22}{3}+\left(\dfrac{51}{2}\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{275}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{231}{5}-\dfrac{43}{2}+\left(-79\right)\sqrt{7}+\left(-1\right)\sqrt{7}\right)-\left(\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{75}{2}\right)\sqrt{7}+\left(-8\right)\sqrt{7}-\dfrac{22}{3}+\left(\dfrac{255}{2}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{425}{3}\right)\sqrt{7}-\dfrac{677}{10}\right)-\left(-\dfrac{29}{2}+\left(-82\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{425}{3}\right)\sqrt{7}-\dfrac{677}{10}+\dfrac{29}{2}+\left(82\right)\sqrt{7}\\
&=&\left(-\dfrac{179}{3}\right)\sqrt{7}-\dfrac{266}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(8\right)\sqrt{63}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{175}\right)-\left(\left(-3\right)\sqrt{28}\right)+\left(-\dfrac{33}{5}\right)\sqrt{49}-\dfrac{43}{2}+\left(-\dfrac{79}{2}\right)\sqrt{28}+\left(-\dfrac{1}{3}\right)\sqrt{63}\right)\times\left(\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{25}{2}\right)\sqrt{63}+\left(-4\right)\sqrt{28}-\dfrac{22}{3}+\left(\dfrac{51}{2}\right)\sqrt{175}\right)\right)\\
&=&\left(\left(\left(24\right)\sqrt{7}\right)-\left(\left(\dfrac{275}{3}\right)\sqrt{7}\right)-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{231}{5}-\dfrac{43}{2}+\left(-79\right)\sqrt{7}+\left(-1\right)\sqrt{7}\right)\times\left(\left(8+\dfrac{4}{3}-\dfrac{63}{2}\right)+\dfrac{1}{3}-\left(\left(-\dfrac{75}{2}\right)\sqrt{7}+\left(-8\right)\sqrt{7}-\dfrac{22}{3}+\left(\dfrac{255}{2}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{425}{3}\right)\sqrt{7}-\dfrac{677}{10}\right)\left(-\dfrac{29}{2}+\left(-82\right)\sqrt{7}\right)\\
&=&\left(\dfrac{228167}{30}\right)\sqrt{7}+\left(\dfrac{34850}{3}\right)\sqrt{49}+\dfrac{19633}{20}\\
&=&\left(\dfrac{228167}{30}\right)\sqrt{7}+\dfrac{4937899}{60}\\
\end{eqnarray*}