Упр.45 Повторение ГДЗ Колмогоров 10-11 класс (Алгебра)
Упростите выражения:
а) $$\left(\frac{3}{2x-y}-\frac{2}{2x+y}-\frac{1}{2x-5y}\right):\frac{4y^2}{4x^2-y^2}$$
б) $$\left(\frac{3}{a-3}+\frac{4}{a^2-5a+6}+\frac{2a}{a-2}\right):\left(\frac{3}{2a+1}\right)^{-1}-\frac{a-12}{3(3-a)}$$
в) $$\left(\frac{x^3-8}{x-2}+2x\right)\cdot(4-x^2)^{-1}-\frac{x-1}{2-x}$$
г) $$\frac{k^2}{3+k}\cdot\frac{9-k^2}{k^2-3k}+\frac{27+k^3}{3-k}:\left(3+\frac{k^2}{3-k}\right)$$
а)
$$\left(\frac{3}{2x-y}-\frac{2}{2x+y}-\frac{1}{2x-5y}\right):\frac{4y^2}{4x^2-y^2}$$
$$=\left(\frac{6x+3y-4x+2y}{(2x-y)(2x+y)}-\frac{1}{2x-5y}\right)\cdot\frac{(2x-y)(2x+y)}{4y^2}$$
$$=\left(\frac{2x+5y}{(2x-y)(2x+y)}-\frac{1}{2x-5y}\right)\cdot\frac{(2x-y)(2x+y)}{4y^2}$$
$$=\frac{(2x+5y)(2x-5y)-(2x-y)(2x+y)}{(2x-5y)\cdot 4y^2}$$
$$=\frac{4x^2-25y^2-4x^2+y^2}{(2x-5y)\cdot 4y^2} =\frac{-24y^2}{4y^2(2x-5y)} =\frac{6}{5y-2x}.$$
б)
$$\left(\frac{3}{a-3}+\frac{4}{a^2-5a+6}+\frac{2a}{a-2}\right):\left(\frac{3}{2a+1}\right)^{-1}-\frac{a-12}{3(3-a)}$$
$$=\left(\frac{3}{a-3}+\frac{4}{(a-3)(a-2)}+\frac{2a}{a-2}\right)\cdot\frac{3}{2a+1}-\frac{a-12}{3(3-a)}$$
$$=\frac{3a-6+4+2a^2-6a}{(a-2)(a-3)}\cdot\frac{3}{2a+1}-\frac{a-12}{3(3-a)}$$
$$=\frac{2a^2-3a-2}{(a-2)(a-3)}\cdot\frac{3}{2a+1}-\frac{a-12}{3(3-a)}$$
$$=\frac{(2a+1)(a-2)}{(a-2)(a-3)}\cdot\frac{3}{2a+1}-\frac{a-12}{3(3-a)}$$
$$=\frac{3}{a-3}-\frac{a-12}{3(3-a)} =\frac{3}{a-3}+\frac{a-12}{3(a-3)} =\frac{9+a-12}{3(a-3)} =\frac{1}{3}.$$
в)
$$\left(\frac{x^3-8}{x-2}+2x\right)\cdot(4-x^2)^{-1}-\frac{x-1}{2-x}$$
$$=\left(\frac{(x-2)(x^2+2x+4)}{x-2}+2x\right)\cdot\frac{1}{4-x^2}-\frac{x-1}{2-x}$$
$$=\frac{x^2+2x+4+2x}{4-x^2}-\frac{x-1}{2-x} =\frac{x^2+4x+4}{(2-x)(2+x)}-\frac{x-1}{2-x}$$
$$=\frac{x+2}{2-x}-\frac{x-1}{2-x} =\frac{x+2-x+1}{2-x} =\frac{3}{2-x}.$$
г)
$$\frac{k^2}{3+k}\cdot\frac{9-k^2}{k^2-3k}+\frac{27+k^3}{3-k}:\left(3+\frac{k^2}{3-k}\right)$$
$$=\frac{k^2}{3+k}\cdot\frac{(3-k)(3+k)}{k(k-3)}+\frac{27+k^3}{3-k}:\frac{9-3k+k^2}{3-k}$$
$$=-k+\frac{27+k^3}{9-3k+k^2} =-k+\frac{(3+k)(9-3k+k^2)}{9-3k+k^2} =-k+3+k =3.$$
Ответ: а) $$\frac{6}{5y-2x}$$; б) $$\frac13$$; в) $$\frac{3}{2-x}$$; г) $$3$$.
