L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{69}{5}\right)\sqrt{12}\right)+\left(-\dfrac{23}{4}\right)\sqrt{9}+\left(-\dfrac{35}{6}\right)\sqrt{12}\) et \( Y=\left(-3\right)\sqrt{75}+\left(-\dfrac{43}{6}\right)\sqrt{12}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(-\dfrac{37}{5}\right)\sqrt{9}+\left(4\right)\sqrt{75}\) . 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{15}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{69}{5}\right)\sqrt{12}\right)+\left(-\dfrac{23}{4}\right)\sqrt{9}+\left(-\dfrac{35}{6}\right)\sqrt{12}\right)+\left(\left(-3\right)\sqrt{75}+\left(-\dfrac{43}{6}\right)\sqrt{12}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(-\dfrac{37}{5}\right)\sqrt{9}+\left(4\right)\sqrt{75}\right)\\
&=&\left(\dfrac{45}{2}-\left(\left(\dfrac{138}{5}\right)\sqrt{3}\right)-\dfrac{69}{4}+\left(-\dfrac{35}{3}\right)\sqrt{3}\right)+\left(\left(-15\right)\sqrt{3}+\left(-\dfrac{43}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{111}{5}+\left(20\right)\sqrt{3}\right)\\
&=&\dfrac{45}{2}-\left(\left(\dfrac{138}{5}\right)\sqrt{3}\right)-\dfrac{69}{4}+\left(-\dfrac{35}{3}\right)\sqrt{3}+\left(-15\right)\sqrt{3}+\left(-\dfrac{43}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{111}{5}+\left(20\right)\sqrt{3}\\
&=&-\dfrac{489}{20}+\left(-\dfrac{243}{5}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{69}{5}\right)\sqrt{12}\right)+\left(-\dfrac{23}{4}\right)\sqrt{9}+\left(-\dfrac{35}{6}\right)\sqrt{12}\right)-\left(\left(-3\right)\sqrt{75}+\left(-\dfrac{43}{6}\right)\sqrt{12}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(-\dfrac{37}{5}\right)\sqrt{9}+\left(4\right)\sqrt{75}\right)\\
&=&\left(\dfrac{45}{2}-\left(\left(\dfrac{138}{5}\right)\sqrt{3}\right)-\dfrac{69}{4}+\left(-\dfrac{35}{3}\right)\sqrt{3}\right)-\left(\left(-15\right)\sqrt{3}+\left(-\dfrac{43}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{111}{5}+\left(20\right)\sqrt{3}\right)\\
&=&\left(\dfrac{21}{4}+\left(-\dfrac{589}{15}\right)\sqrt{3}\right)-\left(\left(-\dfrac{28}{3}\right)\sqrt{3}-\dfrac{297}{10}\right)\\
&=&\dfrac{21}{4}+\left(-\dfrac{589}{15}\right)\sqrt{3}+\left(\dfrac{28}{3}\right)\sqrt{3}+\dfrac{297}{10}\\
&=&\dfrac{699}{20}+\left(-\dfrac{449}{15}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\dfrac{69}{5}\right)\sqrt{12}\right)+\left(-\dfrac{23}{4}\right)\sqrt{9}+\left(-\dfrac{35}{6}\right)\sqrt{12}\right)\times\left(\left(-3\right)\sqrt{75}+\left(-\dfrac{43}{6}\right)\sqrt{12}+\left(-\dfrac{5}{2}\right)\sqrt{9}+\left(-\dfrac{37}{5}\right)\sqrt{9}+\left(4\right)\sqrt{75}\right)\\
&=&\left(\dfrac{45}{2}-\left(\left(\dfrac{138}{5}\right)\sqrt{3}\right)-\dfrac{69}{4}+\left(-\dfrac{35}{3}\right)\sqrt{3}\right)\times\left(\left(-15\right)\sqrt{3}+\left(-\dfrac{43}{3}\right)\sqrt{3}-\dfrac{15}{2}-\dfrac{111}{5}+\left(20\right)\sqrt{3}\right)\\
&=&\left(\dfrac{21}{4}+\left(-\dfrac{589}{15}\right)\sqrt{3}\right)\left(\left(-\dfrac{28}{3}\right)\sqrt{3}-\dfrac{297}{10}\right)\\
&=&\left(\dfrac{55861}{50}\right)\sqrt{3}-\dfrac{6237}{40}+\left(\dfrac{16492}{45}\right)\sqrt{9}\\
&=&\left(\dfrac{55861}{50}\right)\sqrt{3}+\dfrac{22645}{24}\\
\end{eqnarray*}