L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{80}{3}\right)\sqrt{28}\) et \( Y=\left(\left(-5\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{73}{9}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{28}\right)\right)-\left(\left(-8\right)\sqrt{28}+\dfrac{43}{7}+\left(\dfrac{18}{7}\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{63}+\left(\dfrac{39}{4}\right)\sqrt{63}\right)\) . 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{80}{3}\right)\sqrt{28}\right)+\left(\left(\left(-5\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{73}{9}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{28}\right)\right)-\left(\left(-8\right)\sqrt{28}+\dfrac{43}{7}+\left(\dfrac{18}{7}\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{63}+\left(\dfrac{39}{4}\right)\sqrt{63}\right)\right)\\
&=&\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)+\left(\left(\left(-25\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{146}{9}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{77}{2}\right)\sqrt{7}\right)\right)-\left(\left(-16\right)\sqrt{7}+\dfrac{43}{7}+\left(\dfrac{90}{7}\right)\sqrt{7}+\left(\dfrac{45}{2}\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{160}{3}\right)\sqrt{7}+\left(\left(-25\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{146}{9}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{77}{2}\right)\sqrt{7}\right)\right)-\left(\left(-16\right)\sqrt{7}+\dfrac{43}{7}+\left(\dfrac{90}{7}\right)\sqrt{7}+\left(\dfrac{45}{2}\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{42013}{252}\right)\sqrt{7}-\dfrac{43}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{80}{3}\right)\sqrt{28}\right)-\left(\left(\left(-5\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{73}{9}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{28}\right)\right)-\left(\left(-8\right)\sqrt{28}+\dfrac{43}{7}+\left(\dfrac{18}{7}\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{63}+\left(\dfrac{39}{4}\right)\sqrt{63}\right)\right)\\
&=&\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(\left(-25\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{146}{9}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{77}{2}\right)\sqrt{7}\right)\right)-\left(\left(-16\right)\sqrt{7}+\dfrac{43}{7}+\left(\dfrac{90}{7}\right)\sqrt{7}+\left(\dfrac{45}{2}\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{28573}{252}\right)\sqrt{7}-\dfrac{43}{7}\right)\\
&=&\left(-\dfrac{160}{3}\right)\sqrt{7}+\left(\dfrac{28573}{252}\right)\sqrt{7}+\dfrac{43}{7}\\
&=&\left(\dfrac{15133}{252}\right)\sqrt{7}+\dfrac{43}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{80}{3}\right)\sqrt{28}\right)\times\left(\left(\left(-5\right)\sqrt{175}\right)-\left(\left(\left(-\dfrac{73}{9}\right)\sqrt{28}\right)-\left(\left(-9\right)\sqrt{175}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{77}{4}\right)\sqrt{28}\right)\right)-\left(\left(-8\right)\sqrt{28}+\dfrac{43}{7}+\left(\dfrac{18}{7}\right)\sqrt{175}+\left(\dfrac{15}{2}\right)\sqrt{63}+\left(\dfrac{39}{4}\right)\sqrt{63}\right)\right)\\
&=&\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)\times\left(\left(\left(-25\right)\sqrt{7}\right)-\left(\left(\left(-\dfrac{146}{9}\right)\sqrt{7}\right)-\left(\left(-45\right)\sqrt{7}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{77}{2}\right)\sqrt{7}\right)\right)-\left(\left(-16\right)\sqrt{7}+\dfrac{43}{7}+\left(\dfrac{90}{7}\right)\sqrt{7}+\left(\dfrac{45}{2}\right)\sqrt{7}+\left(\dfrac{117}{4}\right)\sqrt{7}\right)\right)\\
&=&\left(\left(-\dfrac{160}{3}\right)\sqrt{7}\right)\left(\left(-\dfrac{28573}{252}\right)\sqrt{7}-\dfrac{43}{7}\right)\\
&=&\left(\dfrac{1142920}{189}\right)\sqrt{49}+\left(\dfrac{6880}{21}\right)\sqrt{7}\\
&=&\dfrac{1142920}{27}+\left(\dfrac{6880}{21}\right)\sqrt{7}\\
\end{eqnarray*}