L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{13}{8}\right)\sqrt{20}\) et \( Y=\left(\dfrac{13}{9}\right)\sqrt{45}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{38}{3}\right)\sqrt{20}-3+\left(-\dfrac{16}{3}\right)\sqrt{20}+\left(-\dfrac{64}{3}\right)\sqrt{125}\) . 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{13}{8}\right)\sqrt{20}\right)+\left(\left(\dfrac{13}{9}\right)\sqrt{45}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{38}{3}\right)\sqrt{20}-3+\left(-\dfrac{16}{3}\right)\sqrt{20}+\left(-\dfrac{64}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{13}{4}\right)\sqrt{5}\right)+\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{76}{3}\right)\sqrt{5}-3+\left(-\dfrac{32}{3}\right)\sqrt{5}+\left(-\dfrac{320}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{13}{4}\right)\sqrt{5}+\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{76}{3}\right)\sqrt{5}-3+\left(-\dfrac{32}{3}\right)\sqrt{5}+\left(-\dfrac{320}{3}\right)\sqrt{5}\\
&=&\left(-\dfrac{1621}{12}\right)\sqrt{5}-\dfrac{29}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{13}{8}\right)\sqrt{20}\right)-\left(\left(\dfrac{13}{9}\right)\sqrt{45}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{38}{3}\right)\sqrt{20}-3+\left(-\dfrac{16}{3}\right)\sqrt{20}+\left(-\dfrac{64}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{13}{4}\right)\sqrt{5}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{76}{3}\right)\sqrt{5}-3+\left(-\dfrac{32}{3}\right)\sqrt{5}+\left(-\dfrac{320}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{13}{4}\right)\sqrt{5}\right)-\left(\left(-\dfrac{415}{3}\right)\sqrt{5}-\dfrac{29}{4}\right)\\
&=&\left(\dfrac{13}{4}\right)\sqrt{5}+\left(\dfrac{415}{3}\right)\sqrt{5}+\dfrac{29}{4}\\
&=&\left(\dfrac{1699}{12}\right)\sqrt{5}+\dfrac{29}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{13}{8}\right)\sqrt{20}\right)\times\left(\left(\dfrac{13}{9}\right)\sqrt{45}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{38}{3}\right)\sqrt{20}-3+\left(-\dfrac{16}{3}\right)\sqrt{20}+\left(-\dfrac{64}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(\dfrac{13}{4}\right)\sqrt{5}\right)\times\left(\left(\dfrac{13}{3}\right)\sqrt{5}-\dfrac{39}{2}+\dfrac{61}{4}+\left(-\dfrac{76}{3}\right)\sqrt{5}-3+\left(-\dfrac{32}{3}\right)\sqrt{5}+\left(-\dfrac{320}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{13}{4}\right)\sqrt{5}\right)\left(\left(-\dfrac{415}{3}\right)\sqrt{5}-\dfrac{29}{4}\right)\\
&=&\left(-\dfrac{5395}{12}\right)\sqrt{25}+\left(-\dfrac{377}{16}\right)\sqrt{5}\\
&=&-\dfrac{26975}{12}+\left(-\dfrac{377}{16}\right)\sqrt{5}\\
\end{eqnarray*}