L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{49}\right)-\left(\dfrac{26}{7}+\left(4\right)\sqrt{63}+\left(-\dfrac{1}{8}\right)\sqrt{49}\right)\) et \( Y=\left(\left(\dfrac{4}{9}\right)\sqrt{63}+\left(4\right)\sqrt{175}+\left(-5\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{63}\right)\right)\) . 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(-\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{49}\right)-\left(\dfrac{26}{7}+\left(4\right)\sqrt{63}+\left(-\dfrac{1}{8}\right)\sqrt{49}\right)\right)+\left(\left(\left(\dfrac{4}{9}\right)\sqrt{63}+\left(4\right)\sqrt{175}+\left(-5\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{63}\right)\right)\right)\\
&=&\left(-\dfrac{161}{3}+\dfrac{70}{3}-\left(\dfrac{26}{7}+\left(12\right)\sqrt{7}-\dfrac{7}{8}\right)\right)+\left(\left(\left(\dfrac{4}{3}\right)\sqrt{7}+\left(20\right)\sqrt{7}+\left(-25\right)\sqrt{7}+\left(\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{7}\right)\right)\right)\\
&=&-\dfrac{161}{3}+\dfrac{70}{3}-\left(\dfrac{26}{7}+\left(12\right)\sqrt{7}-\dfrac{7}{8}\right)+\left(\left(\dfrac{4}{3}\right)\sqrt{7}+\left(20\right)\sqrt{7}+\left(-25\right)\sqrt{7}+\left(\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{7}\right)\right)\\
&=&-\dfrac{5573}{168}+\left(\dfrac{427}{12}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{49}\right)-\left(\dfrac{26}{7}+\left(4\right)\sqrt{63}+\left(-\dfrac{1}{8}\right)\sqrt{49}\right)\right)-\left(\left(\left(\dfrac{4}{9}\right)\sqrt{63}+\left(4\right)\sqrt{175}+\left(-5\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{63}\right)\right)\right)\\
&=&\left(-\dfrac{161}{3}+\dfrac{70}{3}-\left(\dfrac{26}{7}+\left(12\right)\sqrt{7}-\dfrac{7}{8}\right)\right)-\left(\left(\left(\dfrac{4}{3}\right)\sqrt{7}+\left(20\right)\sqrt{7}+\left(-25\right)\sqrt{7}+\left(\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(-\dfrac{5573}{168}+\left(-12\right)\sqrt{7}\right)-\left(\left(\dfrac{571}{12}\right)\sqrt{7}\right)\\
&=&-\dfrac{5573}{168}+\left(-12\right)\sqrt{7}+\left(-\dfrac{571}{12}\right)\sqrt{7}\\
&=&-\dfrac{5573}{168}+\left(-\dfrac{715}{12}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{23}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{49}\right)-\left(\dfrac{26}{7}+\left(4\right)\sqrt{63}+\left(-\dfrac{1}{8}\right)\sqrt{49}\right)\right)\times\left(\left(\left(\dfrac{4}{9}\right)\sqrt{63}+\left(4\right)\sqrt{175}+\left(-5\right)\sqrt{175}+\left(\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{28}\right)-\left(\left(\dfrac{1}{4}\right)\sqrt{63}\right)\right)\right)\\
&=&\left(-\dfrac{161}{3}+\dfrac{70}{3}-\left(\dfrac{26}{7}+\left(12\right)\sqrt{7}-\dfrac{7}{8}\right)\right)\times\left(\left(\left(\dfrac{4}{3}\right)\sqrt{7}+\left(20\right)\sqrt{7}+\left(-25\right)\sqrt{7}+\left(\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(-\dfrac{5573}{168}+\left(-12\right)\sqrt{7}\right)\left(\left(\dfrac{571}{12}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{3182183}{2016}\right)\sqrt{7}+\left(-571\right)\sqrt{49}\\
&=&\left(-\dfrac{3182183}{2016}\right)\sqrt{7}-3997\\
\end{eqnarray*}