L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{62}{7}\right)\sqrt{27}\) et \( Y=\left(\left(5\right)\sqrt{75}+\left(-\dfrac{14}{5}\right)\sqrt{9}+\left(-4\right)\sqrt{9}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{27}-\dfrac{17}{2}+\left(\dfrac{7}{4}\right)\sqrt{27}+\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{12}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{62}{7}\right)\sqrt{27}\right)+\left(\left(\left(5\right)\sqrt{75}+\left(-\dfrac{14}{5}\right)\sqrt{9}+\left(-4\right)\sqrt{9}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{27}-\dfrac{17}{2}+\left(\dfrac{7}{4}\right)\sqrt{27}+\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{3}\right)+\left(\left(\left(25\right)\sqrt{3}-\dfrac{42}{5}-12\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}-\dfrac{17}{2}+\left(\dfrac{21}{4}\right)\sqrt{3}+3\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{186}{7}\right)\sqrt{3}+\left(\left(25\right)\sqrt{3}-\dfrac{42}{5}-12\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}-\dfrac{17}{2}+\left(\dfrac{21}{4}\right)\sqrt{3}+3\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{773}{84}\right)\sqrt{3}-\dfrac{149}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{62}{7}\right)\sqrt{27}\right)-\left(\left(\left(5\right)\sqrt{75}+\left(-\dfrac{14}{5}\right)\sqrt{9}+\left(-4\right)\sqrt{9}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{27}-\dfrac{17}{2}+\left(\dfrac{7}{4}\right)\sqrt{27}+\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{3}\right)-\left(\left(\left(25\right)\sqrt{3}-\dfrac{42}{5}-12\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}-\dfrac{17}{2}+\left(\dfrac{21}{4}\right)\sqrt{3}+3\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{1459}{84}\right)\sqrt{3}-\dfrac{149}{10}\right)\\
&=&\left(-\dfrac{186}{7}\right)\sqrt{3}+\left(-\dfrac{1459}{84}\right)\sqrt{3}+\dfrac{149}{10}\\
&=&\left(-\dfrac{3691}{84}\right)\sqrt{3}+\dfrac{149}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{62}{7}\right)\sqrt{27}\right)\times\left(\left(\left(5\right)\sqrt{75}+\left(-\dfrac{14}{5}\right)\sqrt{9}+\left(-4\right)\sqrt{9}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{27}-\dfrac{17}{2}+\left(\dfrac{7}{4}\right)\sqrt{27}+\left(1\right)\sqrt{9}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{12}\right)\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{3}\right)\times\left(\left(\left(25\right)\sqrt{3}-\dfrac{42}{5}-12\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}-\dfrac{17}{2}+\left(\dfrac{21}{4}\right)\sqrt{3}+3\right)-\left(\left(\dfrac{26}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{186}{7}\right)\sqrt{3}\right)\left(\left(\dfrac{1459}{84}\right)\sqrt{3}-\dfrac{149}{10}\right)\\
&=&\left(-\dfrac{45229}{98}\right)\sqrt{9}+\left(\dfrac{13857}{35}\right)\sqrt{3}\\
&=&-\dfrac{135687}{98}+\left(\dfrac{13857}{35}\right)\sqrt{3}\\
\end{eqnarray*}