L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{20}{3}\right)\sqrt{27}\right)+\dfrac{63}{5}+\left(-6\right)\sqrt{9}+\left(\dfrac{53}{5}\right)\sqrt{27}+\left(5\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{75}+\left(-6\right)\sqrt{9}+\left(\dfrac{61}{6}\right)\sqrt{75}\) et \( Y=\left(\left(\left(-\dfrac{78}{5}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\dfrac{20}{3}\right)\sqrt{27}\right)+\dfrac{63}{5}+\left(-6\right)\sqrt{9}+\left(\dfrac{53}{5}\right)\sqrt{27}+\left(5\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{75}+\left(-6\right)\sqrt{9}+\left(\dfrac{61}{6}\right)\sqrt{75}\right)+\left(\left(\left(\left(-\dfrac{78}{5}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(20\right)\sqrt{3}\right)+\dfrac{63}{5}-18+\left(\dfrac{159}{5}\right)\sqrt{3}+\left(10\right)\sqrt{3}+\left(\dfrac{295}{3}\right)\sqrt{3}-18+\left(\dfrac{305}{6}\right)\sqrt{3}\right)+\left(\left(-\dfrac{234}{5}+15-\left(\left(\dfrac{105}{2}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-0\right)\\
&=&\left(\left(20\right)\sqrt{3}\right)+\dfrac{63}{5}-18+\left(\dfrac{159}{5}\right)\sqrt{3}+\left(10\right)\sqrt{3}+\left(\dfrac{295}{3}\right)\sqrt{3}-18+\left(\dfrac{305}{6}\right)\sqrt{3}+\left(-\dfrac{234}{5}+15-\left(\left(\dfrac{105}{2}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-0\\
&=&\left(\dfrac{799}{5}\right)\sqrt{3}-\dfrac{276}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{20}{3}\right)\sqrt{27}\right)+\dfrac{63}{5}+\left(-6\right)\sqrt{9}+\left(\dfrac{53}{5}\right)\sqrt{27}+\left(5\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{75}+\left(-6\right)\sqrt{9}+\left(\dfrac{61}{6}\right)\sqrt{75}\right)-\left(\left(\left(\left(-\dfrac{78}{5}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(20\right)\sqrt{3}\right)+\dfrac{63}{5}-18+\left(\dfrac{159}{5}\right)\sqrt{3}+\left(10\right)\sqrt{3}+\left(\dfrac{295}{3}\right)\sqrt{3}-18+\left(\dfrac{305}{6}\right)\sqrt{3}\right)-\left(\left(-\dfrac{234}{5}+15-\left(\left(\dfrac{105}{2}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-0\right)\\
&=&\left(\left(\dfrac{6329}{30}\right)\sqrt{3}-\dfrac{117}{5}\right)-\left(-\dfrac{159}{5}+\left(-\dfrac{307}{6}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{6329}{30}\right)\sqrt{3}-\dfrac{117}{5}+\dfrac{159}{5}+\left(\dfrac{307}{6}\right)\sqrt{3}\\
&=&\left(\dfrac{3932}{15}\right)\sqrt{3}+\dfrac{42}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{20}{3}\right)\sqrt{27}\right)+\dfrac{63}{5}+\left(-6\right)\sqrt{9}+\left(\dfrac{53}{5}\right)\sqrt{27}+\left(5\right)\sqrt{12}+\left(\dfrac{59}{3}\right)\sqrt{75}+\left(-6\right)\sqrt{9}+\left(\dfrac{61}{6}\right)\sqrt{75}\right)\times\left(\left(\left(\left(-\dfrac{78}{5}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{9}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{12}\right)-\left(\left(0\right)\sqrt{9}\right)\right)\\
&=&\left(\left(\left(20\right)\sqrt{3}\right)+\dfrac{63}{5}-18+\left(\dfrac{159}{5}\right)\sqrt{3}+\left(10\right)\sqrt{3}+\left(\dfrac{295}{3}\right)\sqrt{3}-18+\left(\dfrac{305}{6}\right)\sqrt{3}\right)\times\left(\left(-\dfrac{234}{5}+15-\left(\left(\dfrac{105}{2}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{3}\right)-0\right)\\
&=&\left(\left(\dfrac{6329}{30}\right)\sqrt{3}-\dfrac{117}{5}\right)\left(-\dfrac{159}{5}+\left(-\dfrac{307}{6}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{137786}{25}\right)\sqrt{3}+\left(-\dfrac{1943003}{180}\right)\sqrt{9}+\dfrac{18603}{25}\\
&=&\left(-\dfrac{137786}{25}\right)\sqrt{3}-\dfrac{9491779}{300}\\
\end{eqnarray*}