Tính một cách hợp lí! $a)~\frac{x+3y}{3x+y}.\frac{4x-2y}{x-y}-\frac{x+3y}{3x+y}.\frac{x-3y}{x-y}$,$b)~\frac{5x+1}{2x-3}.\frac{x+2}{25x^2-1}-\frac{8-3x}{25x^2-1}:\frac{2x-3}{5x+1}$,$c)~\frac{x^2-81}{x^2+101}.(\frac{x^2+101}{x-9}+\frac{x^2+101}{x+9})$

Question

Accepted Answer

a) $\frac{x+3y}{3x+y}.\frac{4x-2y}{x-y}-\frac{x+3y}{3x+y}.\frac{x-3y}{x-y}$

= $\frac{x+3y}{3x+y}.(\frac{4x-2y}{x-y}-\frac{x-3y}{x-y})$

= $\frac{x+3y}{3x+y}.\frac{(4x-2y)-(x-3y)}{x-y}$

= $\frac{x+3y}{3x+y}.\frac{4x-2y-x+3y}{x-y}$

= $\frac{x+3y}{3x+y}.\frac{3x+y}{x-y}$

= $\frac{x+3y}{x-y}$

b) $\frac{5x+1}{2x-3}.\frac{x+2}{25x^2-1}-\frac{8-3x}{25x^2-1}:\frac{2x-3}{5x+1}$

= $\frac{5x+1}{2x-3}.\frac{x+2}{(5x+1)(5x-1)}-\frac{8-3x}{(5x+1)(5x-1)}.\frac{5x+1}{2x-3}$

= $\frac{5x+1}{2x-3}.(\frac{x+2}{(5x+1)(5x-1)}-\frac{8-3x}{(5x+1)(5x-1)})$

= $\frac{5x+1}{2x-3}.(\frac{x+2-(8-3x)}{(5x+1)(5x-1)})$

= $\frac{5x+1}{2x-3}.(\frac{x+2-8+3x}{(5x+1)(5x-1)})$

= $\frac{5x+1}{2x-3}.(\frac{4x-6}{(5x+1)(5x-1)})$

= $\frac{5x+1}{2x-3}.(\frac{2(2x-3)}{(5x+1)(5x-1)})$

= $\frac{2}{5x-1}$

c) $\frac{x^2-81}{x^2+101}.(\frac{x^2+101}{x-9}+\frac{x^2+101}{x+9})$

= $\frac{x^2-81}{x^2+101}.(\frac{x^2+101}{x-9}+\frac{x^2+101}{x+9})$

= $\frac{(x-9)(x+9)}{x^2+101}.(\frac{x^2+101}{x-9}+\frac{x^2+101}{x+9})$

= $(x-9)(x+9).(\frac{1}{x-9}+\frac{1}{x+9})$

= $(x-9)(x+9).\frac{(x+9)+(x-9)}{(x-9)(x+9)}$

= $(x-9)(x+9).\frac{2x}{(x-9)(x+9)}$

= $2x$