L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-1\right)\sqrt{9}+\left(\dfrac{74}{9}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{9}+\left(\dfrac{26}{7}\right)\sqrt{27}-\dfrac{71}{9}+\left(\left(5\right)\sqrt{9}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)+\dfrac{63}{2}\) et \( Y=\left(-\dfrac{35}{4}\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-1\right)\sqrt{9}+\left(\dfrac{74}{9}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{9}+\left(\dfrac{26}{7}\right)\sqrt{27}-\dfrac{71}{9}+\left(\left(5\right)\sqrt{9}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)+\dfrac{63}{2}\right)+\left(\left(-\dfrac{35}{4}\right)\sqrt{9}\right)\\
&=&\left(-3+\dfrac{74}{3}+20+\left(\dfrac{78}{7}\right)\sqrt{3}-\dfrac{71}{9}+15-\left(\left(\dfrac{76}{3}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)+\dfrac{63}{2}\right)+\left(-\dfrac{105}{4}\right)\\
&=&-3+\dfrac{74}{3}+20+\left(\dfrac{78}{7}\right)\sqrt{3}-\dfrac{71}{9}+15-\left(\left(\dfrac{76}{3}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)+\dfrac{63}{2}-\dfrac{105}{4}\\
&=&\dfrac{1945}{36}+\left(-\dfrac{298}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-1\right)\sqrt{9}+\left(\dfrac{74}{9}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{9}+\left(\dfrac{26}{7}\right)\sqrt{27}-\dfrac{71}{9}+\left(\left(5\right)\sqrt{9}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)+\dfrac{63}{2}\right)-\left(\left(-\dfrac{35}{4}\right)\sqrt{9}\right)\\
&=&\left(-3+\dfrac{74}{3}+20+\left(\dfrac{78}{7}\right)\sqrt{3}-\dfrac{71}{9}+15-\left(\left(\dfrac{76}{3}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)+\dfrac{63}{2}\right)-\left(-\dfrac{105}{4}\right)\\
&=&\left(\dfrac{1445}{18}+\left(-\dfrac{298}{21}\right)\sqrt{3}\right)-\left(-\dfrac{105}{4}\right)\\
&=&\dfrac{1445}{18}+\left(-\dfrac{298}{21}\right)\sqrt{3}+\dfrac{105}{4}\\
&=&\dfrac{3835}{36}+\left(-\dfrac{298}{21}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-1\right)\sqrt{9}+\left(\dfrac{74}{9}\right)\sqrt{9}+\left(\dfrac{20}{3}\right)\sqrt{9}+\left(\dfrac{26}{7}\right)\sqrt{27}-\dfrac{71}{9}+\left(\left(5\right)\sqrt{9}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{75}\right)+\dfrac{63}{2}\right)\times\left(\left(-\dfrac{35}{4}\right)\sqrt{9}\right)\\
&=&\left(-3+\dfrac{74}{3}+20+\left(\dfrac{78}{7}\right)\sqrt{3}-\dfrac{71}{9}+15-\left(\left(\dfrac{76}{3}\right)\sqrt{3}\right)-\left(\left(0\right)\sqrt{3}\right)+\dfrac{63}{2}\right)\times\left(-\dfrac{105}{4}\right)\\
&=&\left(\dfrac{1445}{18}+\left(-\dfrac{298}{21}\right)\sqrt{3}\right)\left(-\dfrac{105}{4}\right)\\
&=&-\dfrac{50575}{24}+\left(\dfrac{745}{2}\right)\sqrt{3}\\
&=&-\dfrac{50575}{24}+\left(\dfrac{745}{2}\right)\sqrt{3}\\
\end{eqnarray*}