L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{69}{8}\right)\sqrt{9}\right)+\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{39}{4}\right)\sqrt{75}-\dfrac{21}{8}+\left(-\dfrac{33}{2}\right)\sqrt{9}+\left(-\dfrac{23}{4}\right)\sqrt{75}\) et \( Y=\left(-\dfrac{56}{9}\right)\sqrt{75}\) . 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{5}{2}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{69}{8}\right)\sqrt{9}\right)+\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{39}{4}\right)\sqrt{75}-\dfrac{21}{8}+\left(-\dfrac{33}{2}\right)\sqrt{9}+\left(-\dfrac{23}{4}\right)\sqrt{75}\right)+\left(\left(-\dfrac{56}{9}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(-25\right)\sqrt{3}\right)-\dfrac{207}{8}+\left(-5\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}-\dfrac{21}{8}-\dfrac{99}{2}+\left(-\dfrac{115}{4}\right)\sqrt{3}\right)+\left(\left(-\dfrac{280}{9}\right)\sqrt{3}\right)\\
&=&\left(\left(-5\right)\sqrt{3}\right)-\left(\left(-25\right)\sqrt{3}\right)-\dfrac{207}{8}+\left(-5\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}-\dfrac{21}{8}-\dfrac{99}{2}+\left(-\dfrac{115}{4}\right)\sqrt{3}+\left(-\dfrac{280}{9}\right)\sqrt{3}\\
&=&\left(\dfrac{35}{9}\right)\sqrt{3}-78\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{69}{8}\right)\sqrt{9}\right)+\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{39}{4}\right)\sqrt{75}-\dfrac{21}{8}+\left(-\dfrac{33}{2}\right)\sqrt{9}+\left(-\dfrac{23}{4}\right)\sqrt{75}\right)-\left(\left(-\dfrac{56}{9}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(-25\right)\sqrt{3}\right)-\dfrac{207}{8}+\left(-5\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}-\dfrac{21}{8}-\dfrac{99}{2}+\left(-\dfrac{115}{4}\right)\sqrt{3}\right)-\left(\left(-\dfrac{280}{9}\right)\sqrt{3}\right)\\
&=&\left(\left(35\right)\sqrt{3}-78\right)-\left(\left(-\dfrac{280}{9}\right)\sqrt{3}\right)\\
&=&\left(35\right)\sqrt{3}-78+\left(\dfrac{280}{9}\right)\sqrt{3}\\
&=&\left(\dfrac{595}{9}\right)\sqrt{3}-78\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-5\right)\sqrt{75}\right)-\left(\left(\dfrac{69}{8}\right)\sqrt{9}\right)+\left(-\dfrac{5}{2}\right)\sqrt{12}+\left(\dfrac{39}{4}\right)\sqrt{75}-\dfrac{21}{8}+\left(-\dfrac{33}{2}\right)\sqrt{9}+\left(-\dfrac{23}{4}\right)\sqrt{75}\right)\times\left(\left(-\dfrac{56}{9}\right)\sqrt{75}\right)\\
&=&\left(\left(\left(-5\right)\sqrt{3}\right)-\left(\left(-25\right)\sqrt{3}\right)-\dfrac{207}{8}+\left(-5\right)\sqrt{3}+\left(\dfrac{195}{4}\right)\sqrt{3}-\dfrac{21}{8}-\dfrac{99}{2}+\left(-\dfrac{115}{4}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{280}{9}\right)\sqrt{3}\right)\\
&=&\left(\left(35\right)\sqrt{3}-78\right)\left(\left(-\dfrac{280}{9}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{9800}{9}\right)\sqrt{9}+\left(\dfrac{7280}{3}\right)\sqrt{3}\\
&=&-\dfrac{9800}{3}+\left(\dfrac{7280}{3}\right)\sqrt{3}\\
\end{eqnarray*}