L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{28}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{20}\right)\) et \( Y=\left(-\dfrac{13}{9}\right)\sqrt{45}+\left(\left(-\dfrac{43}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{125}\right)+\left(0\right)\sqrt{125}+\left(\dfrac{37}{2}\right)\sqrt{25}+\left(-7\right)\sqrt{125}\) . 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{28}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{20}\right)\right)+\left(\left(-\dfrac{13}{9}\right)\sqrt{45}+\left(\left(-\dfrac{43}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{125}\right)+\left(0\right)\sqrt{125}+\left(\dfrac{37}{2}\right)\sqrt{25}+\left(-7\right)\sqrt{125}\right)\\
&=&\left(-28-\dfrac{195}{2}-\left(\left(-\dfrac{134}{3}\right)\sqrt{5}\right)\right)+\left(\left(-\dfrac{13}{3}\right)\sqrt{5}+\left(\left(-\dfrac{43}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{205}{2}\right)\sqrt{5}\right)-\left(\left(-36\right)\sqrt{5}\right)+\left(0\right)\sqrt{5}+\dfrac{185}{2}+\left(-35\right)\sqrt{5}\right)\\
&=&-28-\dfrac{195}{2}-\left(\left(-\dfrac{134}{3}\right)\sqrt{5}\right)+\left(-\dfrac{13}{3}\right)\sqrt{5}+\left(\left(-\dfrac{43}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{205}{2}\right)\sqrt{5}\right)-\left(\left(-36\right)\sqrt{5}\right)+\left(0\right)\sqrt{5}+\dfrac{185}{2}+\left(-35\right)\sqrt{5}\\
&=&-33+\left(\dfrac{259}{2}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{28}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{13}{9}\right)\sqrt{45}+\left(\left(-\dfrac{43}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{125}\right)+\left(0\right)\sqrt{125}+\left(\dfrac{37}{2}\right)\sqrt{25}+\left(-7\right)\sqrt{125}\right)\\
&=&\left(-28-\dfrac{195}{2}-\left(\left(-\dfrac{134}{3}\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{13}{3}\right)\sqrt{5}+\left(\left(-\dfrac{43}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{205}{2}\right)\sqrt{5}\right)-\left(\left(-36\right)\sqrt{5}\right)+\left(0\right)\sqrt{5}+\dfrac{185}{2}+\left(-35\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{251}{2}+\left(\dfrac{134}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{509}{6}\right)\sqrt{5}+\dfrac{185}{2}\right)\\
&=&-\dfrac{251}{2}+\left(\dfrac{134}{3}\right)\sqrt{5}+\left(-\dfrac{509}{6}\right)\sqrt{5}-\dfrac{185}{2}\\
&=&-218+\left(-\dfrac{241}{6}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{28}{5}\right)\sqrt{25}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{67}{3}\right)\sqrt{20}\right)\right)\times\left(\left(-\dfrac{13}{9}\right)\sqrt{45}+\left(\left(-\dfrac{43}{6}\right)\sqrt{20}\right)-\left(\left(-\dfrac{41}{2}\right)\sqrt{125}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{125}\right)+\left(0\right)\sqrt{125}+\left(\dfrac{37}{2}\right)\sqrt{25}+\left(-7\right)\sqrt{125}\right)\\
&=&\left(-28-\dfrac{195}{2}-\left(\left(-\dfrac{134}{3}\right)\sqrt{5}\right)\right)\times\left(\left(-\dfrac{13}{3}\right)\sqrt{5}+\left(\left(-\dfrac{43}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{205}{2}\right)\sqrt{5}\right)-\left(\left(-36\right)\sqrt{5}\right)+\left(0\right)\sqrt{5}+\dfrac{185}{2}+\left(-35\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{251}{2}+\left(\dfrac{134}{3}\right)\sqrt{5}\right)\left(\left(\dfrac{509}{6}\right)\sqrt{5}+\dfrac{185}{2}\right)\\
&=&\left(-\dfrac{78179}{12}\right)\sqrt{5}-\dfrac{46435}{4}+\left(\dfrac{34103}{9}\right)\sqrt{25}\\
&=&\left(-\dfrac{78179}{12}\right)\sqrt{5}+\dfrac{264145}{36}\\
\end{eqnarray*}