第9章 9.15

(1)

$(a^{-1}+b^{-1})^{-1} = \frac{1}{a^{-1}+b^{-1}} = \frac{1}{\frac{1}{a}+\frac{1}{b}} = \frac{ab}{a+b}$

(2)

$(2^{-2}-4^{-1})^2 = 2^{-4} - 2\cdot 2^{-2}\cdot 4^{-1} + 4^{-2} = \frac{1}{16} - \frac{2}{16} + \frac{1}{16} = 0$

(3)

$(5t^{-6})^3 \cdot (4t^{-7}) = 125t^{-18}\cdot 4t^{-7} = \frac{500}{t^{25}}$

(4)

$\frac{2x^{-1}}{x^{-1}+y^{-1}} = 2x^{-1}\cdot\frac{xy}{x+y} = \frac{2y}{x+y}$

(5)

$\frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}} = \frac{\dfrac{y^2-x^2}{x^2y^2}}{\dfrac{y-x}{xy}} = \frac{(y-x)(y+x)}{(y-x)xy} = \frac{y+x}{xy}$

(6)

$\left\{\left(a^{\frac{1}{2}}-a^{-\frac{1}{2}}\right)^2+4\right\}^{\frac{1}{2}} = (a-2+a^{-1}+4)^{\frac{1}{2}} = \left\{\left(a^{\frac{1}{2}}+a^{-\frac{1}{2}}\right)^2\right\}^{\frac{1}{2}} = a^{\frac{1}{2}}+a^{-\frac{1}{2}} = \sqrt{a}+\frac{1}{\sqrt{a}}$