L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(0\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\dfrac{5}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)\) et \( Y=\left(\left(7\right)\sqrt{9}\right)+\dfrac{19}{2}-\left(\left(\dfrac{8}{3}\right)\sqrt{12}+\left(-\dfrac{66}{7}\right)\sqrt{9}+\left(-\dfrac{23}{2}\right)\sqrt{27}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+\left(1\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(-2\right)\sqrt{9}\right)\) . 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(0\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\dfrac{5}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)\right)+\left(\left(\left(7\right)\sqrt{9}\right)+\dfrac{19}{2}-\left(\left(\dfrac{8}{3}\right)\sqrt{12}+\left(-\dfrac{66}{7}\right)\sqrt{9}+\left(-\dfrac{23}{2}\right)\sqrt{27}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+\left(1\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(-2\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\dfrac{25}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)\right)+\left(21+\dfrac{19}{2}-\left(\left(\dfrac{16}{3}\right)\sqrt{3}-\dfrac{198}{7}+\left(-\dfrac{69}{2}\right)\sqrt{3}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+3+\left(-21\right)\sqrt{3}-6\right)\right)\\
&=&\left(\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\dfrac{25}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)+21+\dfrac{19}{2}-\left(\left(\dfrac{16}{3}\right)\sqrt{3}-\dfrac{198}{7}+\left(-\dfrac{69}{2}\right)\sqrt{3}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+3+\left(-21\right)\sqrt{3}-6\right)\\
&=&\left(\dfrac{215}{3}\right)\sqrt{3}+\dfrac{4960}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(0\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\dfrac{5}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)\right)-\left(\left(\left(7\right)\sqrt{9}\right)+\dfrac{19}{2}-\left(\left(\dfrac{8}{3}\right)\sqrt{12}+\left(-\dfrac{66}{7}\right)\sqrt{9}+\left(-\dfrac{23}{2}\right)\sqrt{27}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+\left(1\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(-2\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\dfrac{25}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)\right)-\left(21+\dfrac{19}{2}-\left(\left(\dfrac{16}{3}\right)\sqrt{3}-\dfrac{198}{7}+\left(-\dfrac{69}{2}\right)\sqrt{3}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+3+\left(-21\right)\sqrt{3}-6\right)\right)\\
&=&\left(\left(\dfrac{43}{2}\right)\sqrt{3}\right)-\left(\dfrac{4960}{63}+\left(\dfrac{301}{6}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{43}{2}\right)\sqrt{3}+-\dfrac{4960}{63}+\left(-\dfrac{301}{6}\right)\sqrt{3}\\
&=&\left(-\dfrac{86}{3}\right)\sqrt{3}-\dfrac{4960}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(0\right)\sqrt{75}+\left(-7\right)\sqrt{12}+\left(\dfrac{5}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)\right)\times\left(\left(\left(7\right)\sqrt{9}\right)+\dfrac{19}{2}-\left(\left(\dfrac{8}{3}\right)\sqrt{12}+\left(-\dfrac{66}{7}\right)\sqrt{9}+\left(-\dfrac{23}{2}\right)\sqrt{27}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+\left(1\right)\sqrt{9}+\left(-7\right)\sqrt{27}+\left(-2\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(0\right)\sqrt{3}+\left(-14\right)\sqrt{3}+\left(\dfrac{25}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)\right)\times\left(21+\dfrac{19}{2}-\left(\left(\dfrac{16}{3}\right)\sqrt{3}-\dfrac{198}{7}+\left(-\dfrac{69}{2}\right)\sqrt{3}\right)-\left(-\dfrac{19}{2}-\dfrac{67}{9}+3+\left(-21\right)\sqrt{3}-6\right)\right)\\
&=&\left(\left(\dfrac{43}{2}\right)\sqrt{3}\right)\left(\dfrac{4960}{63}+\left(\dfrac{301}{6}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{106640}{63}\right)\sqrt{3}+\left(\dfrac{12943}{12}\right)\sqrt{9}\\
&=&\left(\dfrac{106640}{63}\right)\sqrt{3}+\dfrac{12943}{4}\\
\end{eqnarray*}