第9章 9.8

(1)

$\left(x^{\frac{1}{2}}+y^{\frac{1}{2}}\right)\left(x^{\frac{1}{2}}-y^{\frac{1}{2}}\right) = x^{\frac{1}{2}+\frac{1}{2}} - y^{\frac{1}{2}+\frac{1}{2}} = x - y$

(2)

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

(3)

$(x+x^{-1})^2 = x^2 + 2x^0 + x^{-2} = x^2 + 2 + \frac{1}{x^2}$

(4)

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

(5)

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

(6)

$a^{\frac{1}{2}}\left(1-a^{\frac{1}{2}}+a^{-\frac{3}{2}}\right) = a^{\frac{1}{2}} - a^1 + a^{-1} = \sqrt{a} - a + \frac{1}{a}$

(7)

$\left(x+2^{\frac{1}{2}}\right)^2 = x^2 + 2\cdot 2^{\frac{1}{2}}\cdot x + 2^1 = x^2 + 2\sqrt{2}x + 2$

(8)

$\left(2x-3^{\frac{1}{2}}\right)\left(x+3^{\frac{1}{2}}\right) = 2x^2 + 3^{\frac{1}{2}}x - 3^1 = 2x^2 + \sqrt{3}x - 3$