L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{69}{8}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{13}{5}\right)\sqrt{25}+3\right)-\left(\left(-3\right)\sqrt{25}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{25}\right)\) et \( Y=\left(\dfrac{23}{2}\right)\sqrt{45}\) . 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{69}{8}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{13}{5}\right)\sqrt{25}+3\right)-\left(\left(-3\right)\sqrt{25}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{25}\right)\right)+\left(\left(\dfrac{23}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\dfrac{207}{8}\right)\sqrt{5}+\left(\dfrac{95}{2}\right)\sqrt{5}-13+3\right)-\left(-15+\left(\dfrac{70}{3}\right)\sqrt{5}\right)+15\right)+\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{207}{8}\right)\sqrt{5}+\left(\dfrac{95}{2}\right)\sqrt{5}-13+3\right)-\left(-15+\left(\dfrac{70}{3}\right)\sqrt{5}\right)+15+\left(\dfrac{69}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{2029}{24}\right)\sqrt{5}+20\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{69}{8}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{13}{5}\right)\sqrt{25}+3\right)-\left(\left(-3\right)\sqrt{25}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{25}\right)\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\dfrac{207}{8}\right)\sqrt{5}+\left(\dfrac{95}{2}\right)\sqrt{5}-13+3\right)-\left(-15+\left(\dfrac{70}{3}\right)\sqrt{5}\right)+15\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{1201}{24}\right)\sqrt{5}+20\right)-\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{1201}{24}\right)\sqrt{5}+20+\left(-\dfrac{69}{2}\right)\sqrt{5}\\
&=&\left(\dfrac{373}{24}\right)\sqrt{5}+20\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{69}{8}\right)\sqrt{45}+\left(\dfrac{19}{2}\right)\sqrt{125}+\left(-\dfrac{13}{5}\right)\sqrt{25}+3\right)-\left(\left(-3\right)\sqrt{25}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-3\right)\sqrt{25}\right)\right)\times\left(\left(\dfrac{23}{2}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\dfrac{207}{8}\right)\sqrt{5}+\left(\dfrac{95}{2}\right)\sqrt{5}-13+3\right)-\left(-15+\left(\dfrac{70}{3}\right)\sqrt{5}\right)+15\right)\times\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{1201}{24}\right)\sqrt{5}+20\right)\left(\left(\dfrac{69}{2}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{27623}{16}\right)\sqrt{25}+\left(690\right)\sqrt{5}\\
&=&\dfrac{138115}{16}+\left(690\right)\sqrt{5}\\
\end{eqnarray*}