L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{3}{4}\) et \( Y=\left(-\dfrac{48}{7}\right)\sqrt{27}+\left(0\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{9}+\left(-\dfrac{1}{2}\right)\sqrt{75}-\dfrac{44}{7}+\left(-\dfrac{35}{9}\right)\sqrt{12}+\left(-\dfrac{11}{7}\right)\sqrt{27}+\left(\left(\dfrac{11}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{48}{7}\right)\sqrt{27}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{9}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{3}{4}\right)+\left(\left(-\dfrac{48}{7}\right)\sqrt{27}+\left(0\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{9}+\left(-\dfrac{1}{2}\right)\sqrt{75}-\dfrac{44}{7}+\left(-\dfrac{35}{9}\right)\sqrt{12}+\left(-\dfrac{11}{7}\right)\sqrt{27}+\left(\left(\dfrac{11}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{48}{7}\right)\sqrt{27}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{9}\right)\right)\\
&=&\left(\dfrac{3}{4}\right)+\left(\left(-\dfrac{144}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}-\dfrac{64}{3}+\left(-\dfrac{5}{2}\right)\sqrt{3}-\dfrac{44}{7}+\left(-\dfrac{70}{9}\right)\sqrt{3}+\left(-\dfrac{33}{7}\right)\sqrt{3}+\left(\left(\dfrac{55}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{144}{7}\right)\sqrt{3}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)-\dfrac{87}{8}\right)\\
&=&\dfrac{3}{4}+\left(-\dfrac{144}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}-\dfrac{64}{3}+\left(-\dfrac{5}{2}\right)\sqrt{3}-\dfrac{44}{7}+\left(-\dfrac{70}{9}\right)\sqrt{3}+\left(-\dfrac{33}{7}\right)\sqrt{3}+\left(\left(\dfrac{55}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{144}{7}\right)\sqrt{3}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)-\dfrac{87}{8}\\
&=&-\dfrac{755}{24}+\left(\dfrac{2665}{126}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{3}{4}\right)-\left(\left(-\dfrac{48}{7}\right)\sqrt{27}+\left(0\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{9}+\left(-\dfrac{1}{2}\right)\sqrt{75}-\dfrac{44}{7}+\left(-\dfrac{35}{9}\right)\sqrt{12}+\left(-\dfrac{11}{7}\right)\sqrt{27}+\left(\left(\dfrac{11}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{48}{7}\right)\sqrt{27}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{9}\right)\right)\\
&=&\left(\dfrac{3}{4}\right)-\left(\left(-\dfrac{144}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}-\dfrac{64}{3}+\left(-\dfrac{5}{2}\right)\sqrt{3}-\dfrac{44}{7}+\left(-\dfrac{70}{9}\right)\sqrt{3}+\left(-\dfrac{33}{7}\right)\sqrt{3}+\left(\left(\dfrac{55}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{144}{7}\right)\sqrt{3}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)-\dfrac{87}{8}\right)\\
&=&\left(\dfrac{3}{4}\right)-\left(\left(\dfrac{2665}{126}\right)\sqrt{3}-\dfrac{773}{24}\right)\\
&=&\dfrac{3}{4}+\left(-\dfrac{2665}{126}\right)\sqrt{3}+\dfrac{773}{24}\\
&=&\dfrac{791}{24}+\left(-\dfrac{2665}{126}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{3}{4}\right)\times\left(\left(-\dfrac{48}{7}\right)\sqrt{27}+\left(0\right)\sqrt{75}+\left(-\dfrac{64}{9}\right)\sqrt{9}+\left(-\dfrac{1}{2}\right)\sqrt{75}-\dfrac{44}{7}+\left(-\dfrac{35}{9}\right)\sqrt{12}+\left(-\dfrac{11}{7}\right)\sqrt{27}+\left(\left(\dfrac{11}{7}\right)\sqrt{75}\right)-\left(\left(-\dfrac{48}{7}\right)\sqrt{27}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{66}{7}\right)\sqrt{27}\right)-\left(\left(\dfrac{29}{8}\right)\sqrt{9}\right)\right)\\
&=&\left(\dfrac{3}{4}\right)\times\left(\left(-\dfrac{144}{7}\right)\sqrt{3}+\left(0\right)\sqrt{3}-\dfrac{64}{3}+\left(-\dfrac{5}{2}\right)\sqrt{3}-\dfrac{44}{7}+\left(-\dfrac{70}{9}\right)\sqrt{3}+\left(-\dfrac{33}{7}\right)\sqrt{3}+\left(\left(\dfrac{55}{7}\right)\sqrt{3}\right)-\left(\left(-\dfrac{144}{7}\right)\sqrt{3}\right)+\dfrac{44}{7}-\left(\left(-\dfrac{198}{7}\right)\sqrt{3}\right)-\dfrac{87}{8}\right)\\
&=&\left(\dfrac{3}{4}\right)\left(\left(\dfrac{2665}{126}\right)\sqrt{3}-\dfrac{773}{24}\right)\\
&=&\left(\dfrac{2665}{168}\right)\sqrt{3}-\dfrac{773}{32}\\
&=&\left(\dfrac{2665}{168}\right)\sqrt{3}-\dfrac{773}{32}\\
\end{eqnarray*}