L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-2\right)\sqrt{12}\right)\) et \( Y=\left(-\dfrac{15}{4}\right)\sqrt{75}+\dfrac{62}{3}\) . 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{39}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-2\right)\sqrt{12}\right)\right)+\left(\left(-\dfrac{15}{4}\right)\sqrt{75}+\dfrac{62}{3}\right)\\
&=&\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-4\right)\sqrt{3}\right)\right)+\left(\left(-\dfrac{75}{4}\right)\sqrt{3}+\dfrac{62}{3}\right)\\
&=&\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-4\right)\sqrt{3}\right)+\left(-\dfrac{75}{4}\right)\sqrt{3}+\dfrac{62}{3}\\
&=&\left(\dfrac{203}{10}\right)\sqrt{3}+\dfrac{62}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-2\right)\sqrt{12}\right)\right)-\left(\left(-\dfrac{15}{4}\right)\sqrt{75}+\dfrac{62}{3}\right)\\
&=&\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-4\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{75}{4}\right)\sqrt{3}+\dfrac{62}{3}\right)\\
&=&\left(\left(\dfrac{781}{20}\right)\sqrt{3}\right)-\left(\left(-\dfrac{75}{4}\right)\sqrt{3}+\dfrac{62}{3}\right)\\
&=&\left(\dfrac{781}{20}\right)\sqrt{3}+\left(\dfrac{75}{4}\right)\sqrt{3}-\dfrac{62}{3}\\
&=&\left(\dfrac{289}{5}\right)\sqrt{3}-\dfrac{62}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{27}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-2\right)\sqrt{12}\right)\right)\times\left(\left(-\dfrac{15}{4}\right)\sqrt{75}+\dfrac{62}{3}\right)\\
&=&\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-4\right)\sqrt{3}\right)\right)\times\left(\left(-\dfrac{75}{4}\right)\sqrt{3}+\dfrac{62}{3}\right)\\
&=&\left(\left(\dfrac{781}{20}\right)\sqrt{3}\right)\left(\left(-\dfrac{75}{4}\right)\sqrt{3}+\dfrac{62}{3}\right)\\
&=&\left(-\dfrac{11715}{16}\right)\sqrt{9}+\left(\dfrac{24211}{30}\right)\sqrt{3}\\
&=&-\dfrac{35145}{16}+\left(\dfrac{24211}{30}\right)\sqrt{3}\\
\end{eqnarray*}