Calcul littéral/Exercices/Fractions

Exercice 4-1

Simplifier les expressions suivantes :

1. ${\frac {a}{bc}}+{\frac {b}{ac}}+{\frac {c}{ab}}$ ;
2. ${\frac {{\frac {1}{a}}-{\frac {1}{b}}}{b-a}}$ ;
3. ${\frac {1}{x-1}}+{\frac {2}{x+2}}$ ;
4. ${\frac {1}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}}-{\frac {1}{\left(x+2\right)\left(x+3\right)}}$.

Solution

1. ${\frac {a}{bc}}+{\frac {b}{ac}}+{\frac {c}{ab}}={\frac {a^{2}+b^{2}+c^{2}}{abc}}$ ;
2. ${\frac {{\frac {1}{a}}-{\frac {1}{b}}}{b-a}}={\frac {\frac {b-a}{ab}}{b-a}}={\frac {1}{ab}}$ ;
3. ${\frac {1}{x-1}}+{\frac {2}{x+2}}={\frac {x+2}{(x-1)(x+2)}}+{\frac {2(x-1)}{(x+2)(x-1)}}={\frac {x+2+2(x-1)}{(x+2)(x-1)}}={\frac {3x}{(x+2)(x-1)}}$ ;
4. ${\frac {1}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}}-{\frac {1}{\left(x+2\right)\left(x+3\right)}}={\frac {1}{\left(x+2\right)\left(x+3\right)}}\left({\frac {1}{x+4}}-1\right)={\frac {1}{\left(x+2\right)\left(x+3\right)}}\;{\frac {\left(-x-3\right)}{x+4}}={\frac {-1}{(x+2)(x+4)}}$.