L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{23}{4}\right)\sqrt{27}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\left(-\dfrac{79}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(6\right)\sqrt{27}\right)-\left(\left(6\right)\sqrt{12}\right)\) et \( Y=\dfrac{67}{7}\) . 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{23}{4}\right)\sqrt{27}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\left(-\dfrac{79}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(6\right)\sqrt{27}\right)-\left(\left(6\right)\sqrt{12}\right)\right)+\left(\dfrac{67}{7}\right)\\
&=&\left(\left(\dfrac{69}{4}\right)\sqrt{3}-\dfrac{165}{7}-\dfrac{237}{4}-\dfrac{243}{7}-\left(\left(18\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)\right)+\left(\dfrac{67}{7}\right)\\
&=&\left(\dfrac{69}{4}\right)\sqrt{3}-\dfrac{165}{7}-\dfrac{237}{4}-\dfrac{243}{7}-\left(\left(18\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)+\dfrac{67}{7}\\
&=&\left(-\dfrac{51}{4}\right)\sqrt{3}-\dfrac{3023}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{23}{4}\right)\sqrt{27}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\left(-\dfrac{79}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(6\right)\sqrt{27}\right)-\left(\left(6\right)\sqrt{12}\right)\right)-\left(\dfrac{67}{7}\right)\\
&=&\left(\left(\dfrac{69}{4}\right)\sqrt{3}-\dfrac{165}{7}-\dfrac{237}{4}-\dfrac{243}{7}-\left(\left(18\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)\right)-\left(\dfrac{67}{7}\right)\\
&=&\left(\left(-\dfrac{51}{4}\right)\sqrt{3}-\dfrac{3291}{28}\right)-\left(\dfrac{67}{7}\right)\\
&=&\left(-\dfrac{51}{4}\right)\sqrt{3}-\dfrac{3291}{28}+-\dfrac{67}{7}\\
&=&\left(-\dfrac{51}{4}\right)\sqrt{3}-\dfrac{3559}{28}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{23}{4}\right)\sqrt{27}+\left(-\dfrac{55}{7}\right)\sqrt{9}+\left(\left(-\dfrac{79}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{81}{7}\right)\sqrt{9}\right)-\left(\left(6\right)\sqrt{27}\right)-\left(\left(6\right)\sqrt{12}\right)\right)\times\left(\dfrac{67}{7}\right)\\
&=&\left(\left(\dfrac{69}{4}\right)\sqrt{3}-\dfrac{165}{7}-\dfrac{237}{4}-\dfrac{243}{7}-\left(\left(18\right)\sqrt{3}\right)-\left(\left(12\right)\sqrt{3}\right)\right)\times\left(\dfrac{67}{7}\right)\\
&=&\left(\left(-\dfrac{51}{4}\right)\sqrt{3}-\dfrac{3291}{28}\right)\left(\dfrac{67}{7}\right)\\
&=&\left(-\dfrac{3417}{28}\right)\sqrt{3}-\dfrac{220497}{196}\\
&=&\left(-\dfrac{3417}{28}\right)\sqrt{3}-\dfrac{220497}{196}\\
\end{eqnarray*}