L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{62}{9}\right)\sqrt{4}\) et \( Y=\left(8+\left(\dfrac{37}{2}\right)\sqrt{8}+\left(-\dfrac{52}{9}\right)\sqrt{50}+\left(\dfrac{37}{4}\right)\sqrt{50}\right)-\left(8-1+\left(-\dfrac{15}{2}\right)\sqrt{4}+\left(\dfrac{3}{4}\right)\sqrt{4}\right)\) . 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}{9}\right)\sqrt{4}\right)+\left(\left(8+\left(\dfrac{37}{2}\right)\sqrt{8}+\left(-\dfrac{52}{9}\right)\sqrt{50}+\left(\dfrac{37}{4}\right)\sqrt{50}\right)-\left(8-1+\left(-\dfrac{15}{2}\right)\sqrt{4}+\left(\dfrac{3}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(-\dfrac{124}{9}\right)+\left(\left(8+\left(37\right)\sqrt{2}+\left(-\dfrac{260}{9}\right)\sqrt{2}+\left(\dfrac{185}{4}\right)\sqrt{2}\right)-\left(8-1-15+\dfrac{3}{2}\right)\right)\\
&=&-\dfrac{124}{9}+\left(8+\left(37\right)\sqrt{2}+\left(-\dfrac{260}{9}\right)\sqrt{2}+\left(\dfrac{185}{4}\right)\sqrt{2}\right)-\left(8-1-15+\dfrac{3}{2}\right)\\
&=&\dfrac{13}{18}+\left(\dfrac{1957}{36}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{62}{9}\right)\sqrt{4}\right)-\left(\left(8+\left(\dfrac{37}{2}\right)\sqrt{8}+\left(-\dfrac{52}{9}\right)\sqrt{50}+\left(\dfrac{37}{4}\right)\sqrt{50}\right)-\left(8-1+\left(-\dfrac{15}{2}\right)\sqrt{4}+\left(\dfrac{3}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(-\dfrac{124}{9}\right)-\left(\left(8+\left(37\right)\sqrt{2}+\left(-\dfrac{260}{9}\right)\sqrt{2}+\left(\dfrac{185}{4}\right)\sqrt{2}\right)-\left(8-1-15+\dfrac{3}{2}\right)\right)\\
&=&\left(-\dfrac{124}{9}\right)-\left(\dfrac{29}{2}+\left(\dfrac{1957}{36}\right)\sqrt{2}\right)\\
&=&-\dfrac{124}{9}+-\dfrac{29}{2}+\left(-\dfrac{1957}{36}\right)\sqrt{2}\\
&=&-\dfrac{509}{18}+\left(-\dfrac{1957}{36}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{62}{9}\right)\sqrt{4}\right)\times\left(\left(8+\left(\dfrac{37}{2}\right)\sqrt{8}+\left(-\dfrac{52}{9}\right)\sqrt{50}+\left(\dfrac{37}{4}\right)\sqrt{50}\right)-\left(8-1+\left(-\dfrac{15}{2}\right)\sqrt{4}+\left(\dfrac{3}{4}\right)\sqrt{4}\right)\right)\\
&=&\left(-\dfrac{124}{9}\right)\times\left(\left(8+\left(37\right)\sqrt{2}+\left(-\dfrac{260}{9}\right)\sqrt{2}+\left(\dfrac{185}{4}\right)\sqrt{2}\right)-\left(8-1-15+\dfrac{3}{2}\right)\right)\\
&=&\left(-\dfrac{124}{9}\right)\left(\dfrac{29}{2}+\left(\dfrac{1957}{36}\right)\sqrt{2}\right)\\
&=&-\dfrac{1798}{9}+\left(-\dfrac{60667}{81}\right)\sqrt{2}\\
&=&-\dfrac{1798}{9}+\left(-\dfrac{60667}{81}\right)\sqrt{2}\\
\end{eqnarray*}