5. Преобразуйте в дробь выражение:

Решение:

1. \( (1+a^{-3})(a+1)^{-2} = (1+\frac{1}{a^3})(\frac{1}{(a+1)^2}) = (\frac{a^3+1}{a^3})(\frac{1}{(a+1)^2}) = \frac{(a+1)(a^2-a+1)}{a^3(a+1)^2} = \frac{a^2-a+1}{a^3(a+1)} \)
2. \( (x^2-y^2):(x^{-1}-y^{-1}) = (x^2-y^2):(\frac{1}{x}-\frac{1}{y}) = (x-y)(x+y):(\frac{y-x}{xy}) = (x-y)(x+y) × \frac{xy}{-(x-y)} = \frac{(x+y)xy}{-(x-y)} = -\frac{xy(x+y)}{x-y} \)
3. \( (\frac{a}{c})^{-1} + (\frac{a}{c})^{-3} = \frac{c}{a} + \frac{c^3}{a^3} = \frac{ca^2+c^3}{a^3} = \frac{c(a^2+c^2)}{a^3} \)
4. \( (\frac{1}{b^{-3}} + \frac{1}{c^{-2}}) (b+c)^{-1} = (b^3+c^2) \frac{1}{b+c} = \frac{b^3+c^2}{b+c} \)

Ответ: 1) \( \frac{a^2-a+1}{a^3(a+1)} \); 2) \( -\frac{xy(x+y)}{x-y} \); 3) \( \frac{c(a^2+c^2)}{a^3} \); 4) \( \frac{b^3+c^2}{b+c} \).