L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)\) et \( Y=\left(-\dfrac{8}{3}\right)\sqrt{9}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(\left(\dfrac{8}{9}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{27}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)\right)+\left(\left(-\dfrac{8}{3}\right)\sqrt{9}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(\left(\dfrac{8}{9}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-0-\left(\left(-15\right)\sqrt{3}\right)\right)+\left(-8+\left(-69\right)\sqrt{3}+\left(-69\right)\sqrt{3}+\left(\left(\dfrac{16}{9}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)-\left(\left(\dfrac{111}{8}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-0-\left(\left(-15\right)\sqrt{3}\right)-8+\left(-69\right)\sqrt{3}+\left(-69\right)\sqrt{3}+\left(\left(\dfrac{16}{9}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)-\left(\left(\dfrac{111}{8}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{10627}{72}\right)\sqrt{3}-8\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{9}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(\left(\dfrac{8}{9}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-0-\left(\left(-15\right)\sqrt{3}\right)\right)-\left(-8+\left(-69\right)\sqrt{3}+\left(-69\right)\sqrt{3}+\left(\left(\dfrac{16}{9}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)-\left(\left(\dfrac{111}{8}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{9}{2}\right)\sqrt{3}+0\right)-\left(-8+\left(-\dfrac{10951}{72}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{9}{2}\right)\sqrt{3}+0+8+\left(\dfrac{10951}{72}\right)\sqrt{3}\\
&=&\left(\dfrac{11275}{72}\right)\sqrt{3}+8\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{27}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{27}\right)\right)\times\left(\left(-\dfrac{8}{3}\right)\sqrt{9}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(-\dfrac{69}{2}\right)\sqrt{12}+\left(\left(\dfrac{8}{9}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)-\left(\left(\dfrac{37}{8}\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(-\dfrac{21}{2}\right)\sqrt{3}\right)-0-\left(\left(-15\right)\sqrt{3}\right)\right)\times\left(-8+\left(-69\right)\sqrt{3}+\left(-69\right)\sqrt{3}+\left(\left(\dfrac{16}{9}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)-\left(\left(\dfrac{111}{8}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{9}{2}\right)\sqrt{3}+0\right)\left(-8+\left(-\dfrac{10951}{72}\right)\sqrt{3}\right)\\
&=&\left(-36\right)\sqrt{3}+\left(-\dfrac{10951}{16}\right)\sqrt{9}+0\\
&=&\left(-36\right)\sqrt{3}-\dfrac{32853}{16}\\
\end{eqnarray*}