解：$(1)$∵最简公分母为$21a^2b$
∴$\frac 1{3ab^2}=\frac {1· 7a}{3ab^2· 7a}=\frac {7a}{21a^2b^2}，$$\frac 2{7a^2b}=\frac {2· 3b}{7a^2b· 3b}=\frac {6b}{21a^2b^2}$
$(2)$∵最简公分母为$x(x+1)(x-1)$
∴$\frac {x+1}{x^2-x}=\frac {(x+1)^2}{x(x-1)(x+1)}，$$\frac {x-1}{x^2+x}=\frac {(x-1)^2}{x(x+1)(x-1)}$
$(3)$∵最简公分母为$4(x+1)(x-2)$
∴$\frac {x+2}{2x+2}=\frac {2(x+2)(x-2)}{4(x+1)(x-2)}，$$\frac {x}{x^2-x-2}=\frac {4x}{4(x+1)(x-2)}，$$\frac 3{8-4x}=-\frac {3(x+1)}{4(x+1)(x-2)}$
