Выполни действия: 1) a) (\frac{2a}{b^2} - \frac{1}{2a}):(\frac{1}{b} + \frac{1}{2a})

1. Выполни действия: 1) a) $\left(\frac{2a}{b^2} - \frac{1}{2a}\right) : \left(\frac{1}{b} + \frac{1}{2a}\right) = \frac{4a^2 - b^2}{2ab^2} : \frac{2a+b}{2ab} = \frac{(2a-b)(2a+b)}{2ab^2} \cdot \frac{2ab}{2a+b} = \frac{2a-b}{b}$. б) $\left(\frac{2m}{2m-1} + 1\right) \cdot \frac{6m-3}{4m^2 - m} = \frac{2m + 2m - 1}{2m-1} \cdot \frac{3(2m-1)}{m(4m-1)} = \frac{4m-1}{2m-1} \cdot \frac{3(2m-1)}{m(4m-1)} = \frac{3}{m}$. в) $\frac{y-3}{y+3} \cdot \left(y + \frac{y^2}{3-y}\right) = \frac{y-3}{y+3} \cdot \frac{y(3-y)+y^2}{3-y} = \frac{y-3}{y+3} \cdot \frac{3y - y^2 + y^2}{3-y} = \frac{y-3}{y+3} \cdot \frac{3y}{3-y} = -\frac{3y}{y+3}$. г) $\left(x - \frac{5x}{x+2}\right) : \frac{x-3}{x+2} = \frac{x(x+2)-5x}{x+2} : \frac{x-3}{x+2} = \frac{x^2 + 2x - 5x}{x+2} \cdot \frac{x+2}{x-3} = \frac{x^2 - 3x}{x+2} \cdot \frac{x+2}{x-3} = \frac{x(x-3)}{x+2} \cdot \frac{x+2}{x-3} = x$. д) $\frac{6x+y}{3x} - \frac{5y^2}{x^2} \cdot \frac{x}{15y} = \frac{6x+y}{3x} - \frac{5y^2 x}{15x^2 y} = \frac{6x+y}{3x} - \frac{y}{3x} = \frac{6x+y-y}{3x} = \frac{6x}{3x} = 2$. 2) a) $\frac{a^2-x^2}{b^2-16} \cdot \frac{b+4}{a-x} + \frac{x}{4-b} = \frac{(a-x)(a+x)}{(b-4)(b+4)} \cdot \frac{b+4}{a-x} + \frac{x}{4-b} = \frac{a+x}{b-4} + \frac{x}{4-b} = \frac{a+x}{b-4} - \frac{x}{b-4} = \frac{a+x-x}{b-4} = \frac{a}{b-4}$. б) $\frac{x-y}{2x+y} + \frac{1}{x-y} \cdot \frac{x^2-y^2}{2x+y} = \frac{x-y}{2x+y} + \frac{(x-y)(x+y)}{(x-y)(2x+y)} = \frac{x-y}{2x+y} + \frac{x+y}{2x+y} = \frac{x-y+x+y}{2x+y} = \frac{2x}{2x+y}$. в) $\left(\frac{2a^2-a}{a^2-a+1} - 2\right) : \left(\frac{1}{a+1} - \frac{a-1}{a^2-a+1}\right) = \frac{2a^2-a - 2(a^2-a+1)}{a^2-a+1} : \frac{a^2-a+1 - (a-1)(a+1)}{(a+1)(a^2-a+1)} = \frac{2a^2-a-2a^2+2a-2}{a^2-a+1} : \frac{a^2-a+1 - (a^2-1)}{(a+1)(a^2-a+1)} = \frac{a-2}{a^2-a+1} : \frac{a^2-a+1 - a^2+1}{(a+1)(a^2-a+1)} = \frac{a-2}{a^2-a+1} : \frac{2-a}{(a+1)(a^2-a+1)} = \frac{a-2}{a^2-a+1} \cdot \frac{(a+1)(a^2-a+1)}{2-a} = -(a+1) = -a-1$.