L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=1-\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}-\left(\left(-3\right)\sqrt{25}\right)-\left(\left(4\right)\sqrt{45}\right)\) et \( Y=\left(-4\right)\sqrt{45}-8+\left(-\dfrac{79}{4}\right)\sqrt{25}+\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{45}\right)+\dfrac{37}{7}+\left(\dfrac{81}{4}\right)\sqrt{125}+\left(7\right)\sqrt{25}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(1-\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}-\left(\left(-3\right)\sqrt{25}\right)-\left(\left(4\right)\sqrt{45}\right)\right)+\left(\left(-4\right)\sqrt{45}-8+\left(-\dfrac{79}{4}\right)\sqrt{25}+\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{45}\right)+\dfrac{37}{7}+\left(\dfrac{81}{4}\right)\sqrt{125}+\left(7\right)\sqrt{25}\right)\\
&=&\left(1-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}+15-\left(\left(12\right)\sqrt{5}\right)\right)+\left(\left(-12\right)\sqrt{5}-8-\dfrac{395}{4}+\left(\dfrac{305}{9}\right)\sqrt{5}+0+\dfrac{145}{3}-\left(\left(-24\right)\sqrt{5}\right)+\dfrac{37}{7}+\left(\dfrac{405}{4}\right)\sqrt{5}+35\right)\\
&=&1-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}+15-\left(\left(12\right)\sqrt{5}\right)+\left(-12\right)\sqrt{5}-8-\dfrac{395}{4}+\left(\dfrac{305}{9}\right)\sqrt{5}+0+\dfrac{145}{3}-\left(\left(-24\right)\sqrt{5}\right)+\dfrac{37}{7}+\left(\dfrac{405}{4}\right)\sqrt{5}+35\\
&=&-\dfrac{35921}{1260}+\left(\dfrac{3185}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(1-\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}-\left(\left(-3\right)\sqrt{25}\right)-\left(\left(4\right)\sqrt{45}\right)\right)-\left(\left(-4\right)\sqrt{45}-8+\left(-\dfrac{79}{4}\right)\sqrt{25}+\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{45}\right)+\dfrac{37}{7}+\left(\dfrac{81}{4}\right)\sqrt{125}+\left(7\right)\sqrt{25}\right)\\
&=&\left(1-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}+15-\left(\left(12\right)\sqrt{5}\right)\right)-\left(\left(-12\right)\sqrt{5}-8-\dfrac{395}{4}+\left(\dfrac{305}{9}\right)\sqrt{5}+0+\dfrac{145}{3}-\left(\left(-24\right)\sqrt{5}\right)+\dfrac{37}{7}+\left(\dfrac{405}{4}\right)\sqrt{5}+35\right)\\
&=&\left(-\dfrac{467}{45}+\left(-\dfrac{176}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{5297}{36}\right)\sqrt{5}-\dfrac{1523}{84}\right)\\
&=&-\dfrac{467}{45}+\left(-\dfrac{176}{3}\right)\sqrt{5}+\left(-\dfrac{5297}{36}\right)\sqrt{5}+\dfrac{1523}{84}\\
&=&\dfrac{9769}{1260}+\left(-\dfrac{7409}{36}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(1-\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}-\left(\left(-3\right)\sqrt{25}\right)-\left(\left(4\right)\sqrt{45}\right)\right)\times\left(\left(-4\right)\sqrt{45}-8+\left(-\dfrac{79}{4}\right)\sqrt{25}+\left(\dfrac{61}{9}\right)\sqrt{125}+\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{29}{3}\right)\sqrt{25}\right)-\left(\left(-8\right)\sqrt{45}\right)+\dfrac{37}{7}+\left(\dfrac{81}{4}\right)\sqrt{125}+\left(7\right)\sqrt{25}\right)\\
&=&\left(1-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\dfrac{58}{5}+\dfrac{23}{9}-\dfrac{52}{3}+15-\left(\left(12\right)\sqrt{5}\right)\right)\times\left(\left(-12\right)\sqrt{5}-8-\dfrac{395}{4}+\left(\dfrac{305}{9}\right)\sqrt{5}+0+\dfrac{145}{3}-\left(\left(-24\right)\sqrt{5}\right)+\dfrac{37}{7}+\left(\dfrac{405}{4}\right)\sqrt{5}+35\right)\\
&=&\left(-\dfrac{467}{45}+\left(-\dfrac{176}{3}\right)\sqrt{5}\right)\left(\left(\dfrac{5297}{36}\right)\sqrt{5}-\dfrac{1523}{84}\right)\\
&=&\left(-\dfrac{5253733}{11340}\right)\sqrt{5}+\dfrac{711241}{3780}+\left(-\dfrac{233068}{27}\right)\sqrt{25}\\
&=&\left(-\dfrac{5253733}{11340}\right)\sqrt{5}-\dfrac{4385781693}{102060}\\
\end{eqnarray*}