L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{78}{5}\right)\sqrt{27}\) et \( Y=\left(-\dfrac{55}{3}\right)\sqrt{75}+\dfrac{37}{8}+\left(\dfrac{5}{7}\right)\sqrt{75}+\left(-\dfrac{43}{2}\right)\sqrt{75}+\left(-\dfrac{23}{2}\right)\sqrt{12}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\) . 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{78}{5}\right)\sqrt{27}\right)+\left(\left(-\dfrac{55}{3}\right)\sqrt{75}+\dfrac{37}{8}+\left(\dfrac{5}{7}\right)\sqrt{75}+\left(-\dfrac{43}{2}\right)\sqrt{75}+\left(-\dfrac{23}{2}\right)\sqrt{12}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\right)\\
&=&\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)+\left(\left(-\dfrac{275}{3}\right)\sqrt{3}+\dfrac{37}{8}+\left(\dfrac{25}{7}\right)\sqrt{3}+\left(-\dfrac{215}{2}\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\right)\\
&=&\left(-\dfrac{234}{5}\right)\sqrt{3}+\left(-\dfrac{275}{3}\right)\sqrt{3}+\dfrac{37}{8}+\left(\dfrac{25}{7}\right)\sqrt{3}+\left(-\dfrac{215}{2}\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\\
&=&\left(-\dfrac{55733}{210}\right)\sqrt{3}+\dfrac{2451}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{75}+\dfrac{37}{8}+\left(\dfrac{5}{7}\right)\sqrt{75}+\left(-\dfrac{43}{2}\right)\sqrt{75}+\left(-\dfrac{23}{2}\right)\sqrt{12}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\right)\\
&=&\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{275}{3}\right)\sqrt{3}+\dfrac{37}{8}+\left(\dfrac{25}{7}\right)\sqrt{3}+\left(-\dfrac{215}{2}\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\right)\\
&=&\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{9181}{42}\right)\sqrt{3}+\dfrac{2451}{56}\right)\\
&=&\left(-\dfrac{234}{5}\right)\sqrt{3}+\left(\dfrac{9181}{42}\right)\sqrt{3}-\dfrac{2451}{56}\\
&=&\left(\dfrac{36077}{210}\right)\sqrt{3}-\dfrac{2451}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{78}{5}\right)\sqrt{27}\right)\times\left(\left(-\dfrac{55}{3}\right)\sqrt{75}+\dfrac{37}{8}+\left(\dfrac{5}{7}\right)\sqrt{75}+\left(-\dfrac{43}{2}\right)\sqrt{75}+\left(-\dfrac{23}{2}\right)\sqrt{12}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\right)\\
&=&\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{275}{3}\right)\sqrt{3}+\dfrac{37}{8}+\left(\dfrac{25}{7}\right)\sqrt{3}+\left(-\dfrac{215}{2}\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{67}{2}+\dfrac{36}{7}+\dfrac{1}{2}\right)\\
&=&\left(\left(-\dfrac{234}{5}\right)\sqrt{3}\right)\left(\left(-\dfrac{9181}{42}\right)\sqrt{3}+\dfrac{2451}{56}\right)\\
&=&\left(\dfrac{358059}{35}\right)\sqrt{9}+\left(-\dfrac{286767}{140}\right)\sqrt{3}\\
&=&\dfrac{1074177}{35}+\left(-\dfrac{286767}{140}\right)\sqrt{3}\\
\end{eqnarray*}