8 кл. Алгебра. ср. (18) Упростите. 1) $$\frac{x}{3} + \frac{x-2}{4} + \frac{y}{4} + \frac{y-5}{5} + \frac{x+2}{6}$$ 2) $$\frac{3}{2x-1} - \frac{y+1}{5} - \frac{1}{3}$$ 3) $$\frac{1}{a} + \frac{a-2}{a^{2}} - \frac{a}{c+3}$$ 4) $$\frac{3a}{b} \cdot \frac{b}{6a} : \frac{2x-4}{a} \cdot \frac{a}{8x}$$ 5) $$\frac{2x^{2}}{y} \cdot \frac{y^{2}}{12a^{3}} : \frac{ab+b^{2}}{b^{2}} : \frac{y}{y^{2}+3y} \cdot \frac{y}{2y+6}$$ 6) $$\frac{4x}{x^{2}-16} : \frac{4x}{x+4}$$ 7) $$\frac{y^{2}-9}{3ay^{2}} \cdot \frac{gy}{y-3}$$ 8) $$\frac{b}{3a^{2}} : \frac{a^{3}}{b^{3}}$$ 9) $$\frac{a^{3}}{2x^{2}} : \frac{cx^{3}}{y^{2}}$$ 10) $$\frac{xy+y^{2}}{a-3b} : \frac{x^{2}-y^{2}}{2a-6b}$$ 11) $$\frac{1}{a+b} + \frac{a-b}{3a-b} \cdot \frac{x^{2}-y^{2}}{2x+y}$$

Это задание по алгебре для 8 класса. Необходимо упростить выражения.

1. $$\frac{x}{3} + \frac{x-2}{4} + \frac{y}{4} + \frac{y-5}{5} + \frac{x+2}{6} = \frac{20x + 15x - 30 + 15y + 12y - 60 + 10x + 20}{60} = \frac{45x + 27y - 70}{60}$$
2. $$\frac{3}{2x-1} - \frac{y+1}{5} - \frac{1}{3} = \frac{45 - (6x-3)(y+1) - (10x-5)}{15(2x-1)} = \frac{45 - (6xy + 6x - 3y - 3) - 10x + 5}{30x - 15} = \frac{53 - 4x - 6xy + 3y}{30x - 15}$$
3. $$\frac{1}{a} + \frac{a-2}{a^{2}} - \frac{a}{c+3} = \frac{a(c+3) + (a-2)(c+3) - a^{3}}{a^{2}(c+3)} = \frac{ac + 3a + ac + 3a - 2c - 6 - a^{3}}{a^{2}(c+3)} = \frac{2ac + 6a - 2c - 6 - a^{3}}{a^{2}(c+3)}$$
4. $$\frac{3a}{b} \cdot \frac{b}{6a} : \frac{2x-4}{a} \cdot \frac{a}{8x} = \frac{3ab}{6ab} : \frac{2x-4}{a} \cdot \frac{a}{8x} = \frac{1}{2} : \frac{a(2x-4)}{8x} = \frac{1}{2} \cdot \frac{8x}{a(2x-4)} = \frac{4x}{a(2x-4)} = \frac{2x}{a(x-2)}$$
5. $$\frac{3a}{b} \cdot \frac{b}{6a} : \frac{2x-4}{a} = \frac{3ab}{6ab} : \frac{2x-4}{a} = \frac{1}{2} \cdot \frac{a}{2x-4} = \frac{a}{4x-8} = \frac{a}{4(x-2)}$$
6. $$\frac{2x^{2}}{y} \cdot \frac{y^{2}}{12a^{3}} : \frac{ab+b^{2}}{b^{2}} : \frac{y}{y^{2}+3y} \cdot \frac{y}{2y+6} = \frac{2x^{2}y^{2}}{12a^{3}y} : \frac{b(a+b)}{b^{2}} : \frac{y}{y(y+3)} \cdot \frac{y}{2(y+3)} = \frac{x^{2}y}{6a^{3}} : \frac{a+b}{b} : \frac{1}{y+3} \cdot \frac{y}{2(y+3)} = \frac{x^{2}y}{6a^{3}} \cdot \frac{b}{a+b} \cdot (y+3) \cdot \frac{2(y+3)}{y} = \frac{2x^{2}b(y+3)^{2}}{6a^{3}(a+b)} = \frac{x^{2}b(y+3)^{2}}{3a^{3}(a+b)}$$
7. $$\frac{y^{2}-9}{3ay^{2}} \cdot \frac{gy}{y-3} = \frac{(y-3)(y+3)}{3ay^{2}} \cdot \frac{gy}{y-3} = \frac{g(y+3)}{3ay}$$
8. $$\frac{4x}{x^{2}-16} : \frac{4x}{x+4} = \frac{4x}{(x-4)(x+4)} \cdot \frac{x+4}{4x} = \frac{1}{x-4}$$
9. $$\frac{b}{3a^{2}} : \frac{a^{3}}{b^{3}} = \frac{b}{3a^{2}} \cdot \frac{b^{3}}{a^{3}} = \frac{b^{4}}{3a^{5}}$$
10. $$\frac{a^{3}}{2x^{2}} : \frac{cx^{3}}{y^{2}} = \frac{a^{3}}{2x^{2}} \cdot \frac{y^{2}}{cx^{3}} = \frac{a^{3}y^{2}}{2cx^{5}}$$
11. $$\frac{xy+y^{2}}{a-3b} : \frac{x^{2}-y^{2}}{2a-6b} = \frac{y(x+y)}{a-3b} : \frac{(x-y)(x+y)}{2(a-3b)} = \frac{y(x+y)}{a-3b} \cdot \frac{2(a-3b)}{(x-y)(x+y)} = \frac{2y}{x-y}$$
12. $$\frac{1}{a+b} + \frac{a-b}{3a-b} \cdot \frac{x^{2}-y^{2}}{2x+y}$$ - для решения данного примера нужно больше информации.