L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{58}{9}\right)\sqrt{18}\) et \( Y=\left(-7\right)\sqrt{8}+\dfrac{37}{2}+\left(-\dfrac{45}{8}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{8}+\left(\dfrac{29}{2}\right)\sqrt{18}+\left(-\dfrac{45}{2}\right)\sqrt{4}\) . 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{58}{9}\right)\sqrt{18}\right)+\left(\left(-7\right)\sqrt{8}+\dfrac{37}{2}+\left(-\dfrac{45}{8}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{8}+\left(\dfrac{29}{2}\right)\sqrt{18}+\left(-\dfrac{45}{2}\right)\sqrt{4}\right)\\
&=&\left(\left(\dfrac{58}{3}\right)\sqrt{2}\right)+\left(\left(-14\right)\sqrt{2}+\dfrac{37}{2}-\dfrac{45}{4}+\left(-\dfrac{42}{5}\right)\sqrt{2}+\left(\dfrac{87}{2}\right)\sqrt{2}-45\right)\\
&=&\left(\dfrac{58}{3}\right)\sqrt{2}+\left(-14\right)\sqrt{2}+\dfrac{37}{2}-\dfrac{45}{4}+\left(-\dfrac{42}{5}\right)\sqrt{2}+\left(\dfrac{87}{2}\right)\sqrt{2}-45\\
&=&\left(\dfrac{1213}{30}\right)\sqrt{2}-\dfrac{151}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{58}{9}\right)\sqrt{18}\right)-\left(\left(-7\right)\sqrt{8}+\dfrac{37}{2}+\left(-\dfrac{45}{8}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{8}+\left(\dfrac{29}{2}\right)\sqrt{18}+\left(-\dfrac{45}{2}\right)\sqrt{4}\right)\\
&=&\left(\left(\dfrac{58}{3}\right)\sqrt{2}\right)-\left(\left(-14\right)\sqrt{2}+\dfrac{37}{2}-\dfrac{45}{4}+\left(-\dfrac{42}{5}\right)\sqrt{2}+\left(\dfrac{87}{2}\right)\sqrt{2}-45\right)\\
&=&\left(\left(\dfrac{58}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{211}{10}\right)\sqrt{2}-\dfrac{151}{4}\right)\\
&=&\left(\dfrac{58}{3}\right)\sqrt{2}+\left(-\dfrac{211}{10}\right)\sqrt{2}+\dfrac{151}{4}\\
&=&\left(-\dfrac{53}{30}\right)\sqrt{2}+\dfrac{151}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{58}{9}\right)\sqrt{18}\right)\times\left(\left(-7\right)\sqrt{8}+\dfrac{37}{2}+\left(-\dfrac{45}{8}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{8}+\left(\dfrac{29}{2}\right)\sqrt{18}+\left(-\dfrac{45}{2}\right)\sqrt{4}\right)\\
&=&\left(\left(\dfrac{58}{3}\right)\sqrt{2}\right)\times\left(\left(-14\right)\sqrt{2}+\dfrac{37}{2}-\dfrac{45}{4}+\left(-\dfrac{42}{5}\right)\sqrt{2}+\left(\dfrac{87}{2}\right)\sqrt{2}-45\right)\\
&=&\left(\left(\dfrac{58}{3}\right)\sqrt{2}\right)\left(\left(\dfrac{211}{10}\right)\sqrt{2}-\dfrac{151}{4}\right)\\
&=&\left(\dfrac{6119}{15}\right)\sqrt{4}+\left(-\dfrac{4379}{6}\right)\sqrt{2}\\
&=&\dfrac{12238}{15}+\left(-\dfrac{4379}{6}\right)\sqrt{2}\\
\end{eqnarray*}