L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{79}{5}\right)\sqrt{75}\) et \( Y=\left(-\dfrac{41}{6}\right)\sqrt{12}+\left(-1\right)\sqrt{75}+\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(-\dfrac{17}{3}\right)\sqrt{9}+\left(\left(-\dfrac{49}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{5}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}\right)+\left(\dfrac{5}{3}\right)\sqrt{12}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{79}{5}\right)\sqrt{75}\right)+\left(\left(-\dfrac{41}{6}\right)\sqrt{12}+\left(-1\right)\sqrt{75}+\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(-\dfrac{17}{3}\right)\sqrt{9}+\left(\left(-\dfrac{49}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{5}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}\right)+\left(\dfrac{5}{3}\right)\sqrt{12}\right)\\
&=&\left(\left(79\right)\sqrt{3}\right)+\left(\left(-\dfrac{41}{3}\right)\sqrt{3}+\left(-5\right)\sqrt{3}-38-17-\dfrac{147}{8}-\left(\left(\dfrac{129}{5}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)+\left(\dfrac{10}{3}\right)\sqrt{3}\right)\\
&=&\left(79\right)\sqrt{3}+\left(-\dfrac{41}{3}\right)\sqrt{3}+\left(-5\right)\sqrt{3}-38-17-\dfrac{147}{8}-\left(\left(\dfrac{129}{5}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)+\left(\dfrac{10}{3}\right)\sqrt{3}\\
&=&\left(\dfrac{358}{15}\right)\sqrt{3}-\dfrac{587}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{79}{5}\right)\sqrt{75}\right)-\left(\left(-\dfrac{41}{6}\right)\sqrt{12}+\left(-1\right)\sqrt{75}+\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(-\dfrac{17}{3}\right)\sqrt{9}+\left(\left(-\dfrac{49}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{5}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}\right)+\left(\dfrac{5}{3}\right)\sqrt{12}\right)\\
&=&\left(\left(79\right)\sqrt{3}\right)-\left(\left(-\dfrac{41}{3}\right)\sqrt{3}+\left(-5\right)\sqrt{3}-38-17-\dfrac{147}{8}-\left(\left(\dfrac{129}{5}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)+\left(\dfrac{10}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(79\right)\sqrt{3}\right)-\left(\left(-\dfrac{827}{15}\right)\sqrt{3}-\dfrac{587}{8}\right)\\
&=&\left(79\right)\sqrt{3}+\left(\dfrac{827}{15}\right)\sqrt{3}+\dfrac{587}{8}\\
&=&\left(\dfrac{2012}{15}\right)\sqrt{3}+\dfrac{587}{8}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{79}{5}\right)\sqrt{75}\right)\times\left(\left(-\dfrac{41}{6}\right)\sqrt{12}+\left(-1\right)\sqrt{75}+\left(-\dfrac{38}{3}\right)\sqrt{9}+\left(-\dfrac{17}{3}\right)\sqrt{9}+\left(\left(-\dfrac{49}{8}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{5}\right)\sqrt{27}\right)-\left(\left(7\right)\sqrt{12}\right)+\left(\dfrac{5}{3}\right)\sqrt{12}\right)\\
&=&\left(\left(79\right)\sqrt{3}\right)\times\left(\left(-\dfrac{41}{3}\right)\sqrt{3}+\left(-5\right)\sqrt{3}-38-17-\dfrac{147}{8}-\left(\left(\dfrac{129}{5}\right)\sqrt{3}\right)-\left(\left(14\right)\sqrt{3}\right)+\left(\dfrac{10}{3}\right)\sqrt{3}\right)\\
&=&\left(\left(79\right)\sqrt{3}\right)\left(\left(-\dfrac{827}{15}\right)\sqrt{3}-\dfrac{587}{8}\right)\\
&=&\left(-\dfrac{65333}{15}\right)\sqrt{9}+\left(-\dfrac{46373}{8}\right)\sqrt{3}\\
&=&-\dfrac{65333}{5}+\left(-\dfrac{46373}{8}\right)\sqrt{3}\\
\end{eqnarray*}