Упростите выражения с дробными степенями и корнями.

1) $\frac{a - b}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} = \frac{(a^{\frac{1}{2}})^2 - (b^{\frac{1}{2}})^2}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} = \frac{(a^{\frac{1}{2}} - b^{\frac{1}{2}})(a^{\frac{1}{2}} + b^{\frac{1}{2}})}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} = a^{\frac{1}{2}} + b^{\frac{1}{2}}$ 2) $\frac{z - 8}{z^{\frac{2}{3}} + 2z^{\frac{1}{3}} + 4} = \frac{(z^{\frac{1}{3}})^3 - 2^3}{z^{\frac{2}{3}} + 2z^{\frac{1}{3}} + 4} = \frac{(z^{\frac{1}{3}} - 2)(z^{\frac{2}{3}} + 2z^{\frac{1}{3}} + 4)}{z^{\frac{2}{3}} + 2z^{\frac{1}{3}} + 4} = z^{\frac{1}{3}} - 2$ 3) $\frac{x^{\frac{1}{2}} - 4}{x - 16} = \frac{x^{\frac{1}{2}} - 4}{(x^{\frac{1}{2}})^2 - 4^2} = \frac{x^{\frac{1}{2}} - 4}{(x^{\frac{1}{2}} - 4)(x^{\frac{1}{2}} + 4)} = \frac{1}{x^{\frac{1}{2}} + 4}$ 4) $\frac{a + b}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} = \frac{(a^{\frac{1}{3}})^3 + (b^{\frac{1}{3}})^3}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} = \frac{(a^{\frac{1}{3}} + b^{\frac{1}{3}})(a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}})}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} = a^{\frac{1}{3}} + b^{\frac{1}{3}}$ 5) $\frac{x - y}{x^{\frac{3}{4}} + x^{\frac{1}{2}}y^{\frac{1}{4}}} \cdot \frac{x^{\frac{1}{2}}y^{\frac{1}{4}} + x^{\frac{1}{4}}y^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}} = \frac{(x^{\frac{1}{2}} - y^{\frac{1}{2}})(x^{\frac{1}{2}} + y^{\frac{1}{2}})}{x^{\frac{1}{2}}(x^{\frac{1}{4}} + y^{\frac{1}{4}})} \cdot \frac{x^{\frac{1}{4}}y^{\frac{1}{4}}(x^{\frac{1}{4}} + y^{\frac{1}{4}})}{x^{\frac{1}{2}} + y^{\frac{1}{2}}} = \frac{(x^{\frac{1}{4}} - y^{\frac{1}{4}})(x^{\frac{1}{4}} + y^{\frac{1}{4}}) \cdot x^{\frac{1}{4}}y^{\frac{1}{4}}}{x^{\frac{1}{2}}} = \frac{(x^{\frac{1}{4}} - y^{\frac{1}{4}})(x^{\frac{1}{4}} + y^{\frac{1}{4}})x^{\frac{1}{4}}y^{\frac{1}{4}}}{x^{\frac{1}{2}}}$ (после сокращения: $\frac{(x^{\frac{1}{2}} - y^{\frac{1}{2}})x^{\frac{1}{4}}y^{\frac{1}{4}}}{x^{\frac{1}{2}}}$) 6) $\frac{a - 1}{a + a^{\frac{1}{2}} + 1} : \frac{a^{\frac{1}{2}} + 1}{a^{\frac{3}{2}} - 1} + 2a^{\frac{1}{2}} = \frac{(a^{\frac{1}{2}} - 1)(a^{\frac{1}{2}} + 1)}{a + a^{\frac{1}{2}} + 1} \cdot \frac{(a^{\frac{1}{2}} - 1)(a + a^{\frac{1}{2}} + 1)}{a^{\frac{1}{2}} + 1} + 2a^{\frac{1}{2}} = (a^{\frac{1}{2}} - 1)^2 + 2a^{\frac{1}{2}} = a - 2a^{\frac{1}{2}} + 1 + 2a^{\frac{1}{2}} = a + 1$ 7) $(\frac{1}{a + a^{\frac{1}{2}}b^{\frac{1}{2}}} + \frac{1}{a - a^{\frac{1}{2}}b^{\frac{1}{2}}}) \cdot \frac{a^3 - b^3}{a^2 + ab + b^2} = \frac{a - a^{\frac{1}{2}}b^{\frac{1}{2}} + a + a^{\frac{1}{2}}b^{\frac{1}{2}}}{a(a - b)} \cdot (a - b) = \frac{2a}{a(a - b)} \cdot (a - b) = 2$ 8) $\frac{\sqrt{x} + 1}{x\sqrt{x} + x + \sqrt{x}} : \frac{1}{x^2 - \sqrt{x}} = \frac{\sqrt{x} + 1}{\sqrt{x}(x + \sqrt{x} + 1)} \cdot \sqrt{x}(x\sqrt{x} - 1) = \frac{(\sqrt{x} + 1)(\sqrt{x} - 1)(x + \sqrt{x} + 1)}{x + \sqrt{x} + 1} = x - 1$