13. Преобразуйте в дробь выражение: 1) а) (1+a-3)(a+1)-2; б) (x-2-y-2):(x-1-y-1); 2) а) (\(\frac{a}{c}\))⁻¹+(\(\frac{a}{c}\))⁻³; б) \(\frac{1}{b^{-3}+c^{-3}}:(b+c)^{-1}\)

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

1. a) (1+a⁻³)(a+1)⁻² = (1 + \(\frac{1}{a^3}\)) ⋅ \(\frac{1}{(a+1)^2}\) = \(\frac{a^3+1}{a^3}\) ⋅ \(\frac{1}{(a+1)^2}\) = \(\frac{a^3+1}{a^3(a+1)^2}\) b) (x⁻²-y⁻²) : (x⁻¹-y⁻¹) = (\(\frac{1}{x^2}\) - \(\frac{1}{y^2}\)) : (\(\frac{1}{x}\) - \(\frac{1}{y}\)) = (\(\frac{y^2-x^2}{x^2y^2}\)) : (\(\frac{y-x}{xy}\)) = \(\frac{(y-x)(y+x)xy}{x^2y^2(y-x)}\) = \(\frac{y+x}{xy}\)
2. a) (\(\frac{a}{c}\))⁻¹ + (\(\frac{a}{c}\))⁻³ = \(\frac{c}{a}\) + \(\frac{c^3}{a^3}\) b) \(\frac{1}{b^{-3}+c^{-3}}\) : (b+c)⁻¹ = \(\frac{1}{\frac{1}{b^3}+\frac{1}{c^3}}\) : \(\frac{1}{b+c}\) = \(\frac{1}{\frac{c^3+b^3}{b^3c^3}}\) ⋅ (b+c) = \(\frac{b^3c^3}{b^3+c^3}\) ⋅ (b+c) = \(\frac{b^3c^3}{(b+c)(b^2-bc+c^2)}\) ⋅ (b+c) = \(\frac{b^3c^3}{b^2-bc+c^2}\)

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