L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{6}{5}\right)\sqrt{175}\) et \( Y=\left(\left(\left(\dfrac{20}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{175}\right)\right)-\dfrac{7}{8}-\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{49}\right)-\left(\left(\dfrac{11}{7}\right)\sqrt{63}\right)\right)-\left(\left(1\right)\sqrt{28}\right)-\left(\left(\left(1\right)\sqrt{175}\right)-\dfrac{20}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\) . 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{6}{5}\right)\sqrt{175}\right)+\left(\left(\left(\left(\dfrac{20}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{175}\right)\right)-\dfrac{7}{8}-\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{49}\right)-\left(\left(\dfrac{11}{7}\right)\sqrt{63}\right)\right)-\left(\left(1\right)\sqrt{28}\right)-\left(\left(\left(1\right)\sqrt{175}\right)-\dfrac{20}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(6\right)\sqrt{7}\right)+\left(\left(\dfrac{140}{3}-\left(\left(-\dfrac{48}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{7}{8}-\left(-74-\left(\left(\dfrac{33}{7}\right)\sqrt{7}\right)\right)-\left(\left(2\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\dfrac{20}{3}+\dfrac{77}{2}-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\right)\\
&=&\left(6\right)\sqrt{7}+\left(\dfrac{140}{3}-\left(\left(-\dfrac{48}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{7}{8}-\left(-74-\left(\left(\dfrac{33}{7}\right)\sqrt{7}\right)\right)-\left(\left(2\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\dfrac{20}{3}+\dfrac{77}{2}-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{167}{14}\right)\sqrt{7}+\dfrac{2111}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{6}{5}\right)\sqrt{175}\right)-\left(\left(\left(\left(\dfrac{20}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{175}\right)\right)-\dfrac{7}{8}-\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{49}\right)-\left(\left(\dfrac{11}{7}\right)\sqrt{63}\right)\right)-\left(\left(1\right)\sqrt{28}\right)-\left(\left(\left(1\right)\sqrt{175}\right)-\dfrac{20}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(6\right)\sqrt{7}\right)-\left(\left(\dfrac{140}{3}-\left(\left(-\dfrac{48}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{7}{8}-\left(-74-\left(\left(\dfrac{33}{7}\right)\sqrt{7}\right)\right)-\left(\left(2\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\dfrac{20}{3}+\dfrac{77}{2}-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(6\right)\sqrt{7}\right)-\left(\dfrac{2111}{24}+\left(-\dfrac{251}{14}\right)\sqrt{7}\right)\\
&=&\left(6\right)\sqrt{7}+-\dfrac{2111}{24}+\left(\dfrac{251}{14}\right)\sqrt{7}\\
&=&\left(\dfrac{335}{14}\right)\sqrt{7}-\dfrac{2111}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{6}{5}\right)\sqrt{175}\right)\times\left(\left(\left(\left(\dfrac{20}{3}\right)\sqrt{49}\right)-\left(\left(-\dfrac{16}{7}\right)\sqrt{63}\right)-\left(\left(1\right)\sqrt{175}\right)\right)-\dfrac{7}{8}-\left(\left(\left(-\dfrac{74}{7}\right)\sqrt{49}\right)-\left(\left(\dfrac{11}{7}\right)\sqrt{63}\right)\right)-\left(\left(1\right)\sqrt{28}\right)-\left(\left(\left(1\right)\sqrt{175}\right)-\dfrac{20}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)-\left(\left(-7\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(6\right)\sqrt{7}\right)\times\left(\left(\dfrac{140}{3}-\left(\left(-\dfrac{48}{7}\right)\sqrt{7}\right)-\left(\left(5\right)\sqrt{7}\right)\right)-\dfrac{7}{8}-\left(-74-\left(\left(\dfrac{33}{7}\right)\sqrt{7}\right)\right)-\left(\left(2\right)\sqrt{7}\right)-\left(\left(\left(5\right)\sqrt{7}\right)-\dfrac{20}{3}+\dfrac{77}{2}-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)-\left(\left(-35\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(6\right)\sqrt{7}\right)\left(\dfrac{2111}{24}+\left(-\dfrac{251}{14}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{2111}{4}\right)\sqrt{7}+\left(-\dfrac{753}{7}\right)\sqrt{49}\\
&=&\left(\dfrac{2111}{4}\right)\sqrt{7}-753\\
\end{eqnarray*}