Ответы к странице 44

177. Упростите выражение:
1) $(x + \frac{x}{y}) : (x - \frac{x}{y})$;
2) $(\frac{a}{b} + \frac{a + b}{a - b}) * \frac{ab^2}{a^2 + b^2}$;
3) $(\frac{m}{m - 1} - 1) : \frac{m}{mn - n}$;
4) $(\frac{a}{b} - \frac{b}{a}) * \frac{4ab}{a - b}$;
5) $\frac{a}{b} - \frac{a^2 - b^2}{b^2} : \frac{a + b}{b}$;
6) $\frac{7x}{x + 2} - \frac{x - 8}{3x + 6} * \frac{84}{x^2 - 8x}$;
7) $(a - \frac{9a - 9}{a + 3}) : \frac{a^2 - 3a}{a + 3}$;
8) $(\frac{a}{a + 2} - \frac{8}{a + 8}) * \frac{a^2 + 8a}{a - 4}$.

Решение:

1) $(x + \frac{x}{y}) : (x - \frac{x}{y}) = \frac{xy + x}{y} : \frac{xy - x}{y} = \frac{x(y + 1)}{y} : \frac{x(y - 1)}{y} = \frac{x(y + 1)}{y} * \frac{y}{x(y - 1)} = \frac{y + 1}{1} * \frac{1}{y - 1} = \frac{y + 1}{y - 1}$

2) $(\frac{a}{b} + \frac{a + b}{a - b}) * \frac{ab^2}{a^2 + b^2} = \frac{a(a - b) + b(a + b)}{b(a - b)} * \frac{ab^2}{a^2 + b^2} = \frac{a^2 - ab + ab + b^2}{b(a - b)} * \frac{ab^2}{a^2 + b^2} = \frac{a^2 + b^2}{b(a - b)} * \frac{ab^2}{a^2 + b^2} = \frac{1}{a - b} * \frac{ab}{1} = \frac{ab}{a - b}$

3) $(\frac{m}{m - 1} - 1) : \frac{m}{mn - n} = \frac{m - (m - 1)}{m - 1} : \frac{m}{n(m - 1)} = \frac{m - m + 1}{m - 1} : \frac{m}{n(m - 1)} = \frac{1}{m - 1} * \frac{n(m - 1)}{m} = \frac{1}{1} * \frac{n}{m} = \frac{n}{m}$

4) $(\frac{a}{b} - \frac{b}{a}) * \frac{4ab}{a - b} = \frac{a * a - b * b}{ab} * \frac{4ab}{a - b} = \frac{a^2 - b^2}{1} * \frac{4}{a - b} = \frac{(a - b)(a + b)}{1} * \frac{4}{a - b} = \frac{a + b}{1} * \frac{4}{1} = 4(a + b)$

5) $\frac{a}{b} - \frac{a^2 - b^2}{b^2} : \frac{a + b}{b} = \frac{a}{b} - \frac{(a - b)(a + b)}{b^2} * \frac{b}{a + b} = \frac{a}{b} - \frac{a - b}{b} * \frac{1}{1} = \frac{a - (a - b)}{b} = \frac{a - a + b}{b} = \frac{b}{b} = 1$

6) $\frac{7x}{x + 2} - \frac{x - 8}{3x + 6} * \frac{84}{x^2 - 8x} = \frac{7x}{x + 2} - \frac{x - 8}{3(x + 2)} * \frac{84}{x(x - 8)} = \frac{7x}{x + 2} - \frac{1}{x + 2} * \frac{28}{x} = \frac{7x}{x + 2} - \frac{28}{x(x + 2)} = \frac{7x * x - 28}{x(x + 2)} = \frac{7x^2 - 28}{x(x + 2)} = \frac{7(x^2 - 4)}{x(x + 2)} = \frac{7(x - 2)(x + 2)}{x(x + 2)} = \frac{7(x - 2)}{x}$

7) $(a - \frac{9a - 9}{a + 3}) : \frac{a^2 - 3a}{a + 3} = \frac{a(a + 3) - (9a - 9)}{a + 3} : \frac{a(a - 3)}{a + 3} = \frac{a^2 + 3a - 9a + 9}{a + 3} * \frac{a + 3}{a(a - 3)} = \frac{a^2 - 6a + 9}{1} * \frac{1}{a(a - 3)} = \frac{(a - 3)^2}{a(a - 3)} = \frac{a - 3}{a}$

8) $(\frac{a}{a + 2} - \frac{8}{a + 8}) * \frac{a^2 + 8a}{a - 4} = \frac{a(a + 8) - 8(a + 2)}{(a + 2)(a + 8)} * \frac{a(a + 8)}{a - 4} = \frac{a^2 + 8a - 8a - 16}{a + 2} * \frac{a}{a - 4} = \frac{a^2 - 16}{a + 2} * \frac{a}{a - 4} = \frac{(a - 4)(a + 4)}{a + 2} * \frac{a}{a - 4} = \frac{a + 4}{a + 2} * \frac{a}{1} = \frac{a(a + 4)}{a + 2}$

178. Выполните действия:
1) $\frac{a + 2}{a^2 - 2a + 1} : \frac{a^2 - 4}{3a - 3} - \frac{3}{a - 2}$;
2) $\frac{b^2 + 3b}{b^3 + 9b} * (\frac{b - 3}{b + 3} + \frac{b + 3}{b - 3})$;
3) $(\frac{3c + 1}{3c - 1} - \frac{3c - 1}{3c + 1}) : \frac{2c}{6c + 2}$;
4) $(\frac{1}{a^2 - 4ab + 4b^2} - \frac{1}{4b^2 - a^2}) : \frac{2a}{a^2 - 4b^2}$;
5) $(\frac{a - 8}{a^2 - 10a + 25} - \frac{a}{a^2 - 25}) : \frac{a - 20}{(a - 5)^2}$;
6) $(\frac{2x + 1}{x^2 + 6x + 9} - \frac{x - 2}{x^2 + 3x}) : \frac{x^2 + 6}{x^3 - 9x}$.

Решение:

1) $\frac{a + 2}{a^2 - 2a + 1} : \frac{a^2 - 4}{3a - 3} - \frac{3}{a - 2} = \frac{a + 2}{(a - 1)^2} : \frac{(a - 2)(a + 2)}{3(a - 1)} - \frac{3}{a - 2} = \frac{a + 2}{(a - 1)^2} * \frac{3(a - 1)}{(a - 2)(a + 2)} - \frac{3}{a - 2} = \frac{1}{a - 1} * \frac{3}{a - 2} - \frac{3}{a - 2} = \frac{3}{(a - 1)(a - 2)} - \frac{3}{a - 2} = \frac{3 - 3(a - 1)}{(a - 1)(a - 2)} = \frac{3 - 3a + 3}{(a - 1)(a - 2)} = \frac{6 - 3a}{(a - 1)(a - 2)} = \frac{3(2 - a)}{(a - 1)(a - 2)} = -\frac{3(a - 2)}{(a - 1)(a - 2)} = -\frac{3}{a - 1} = \frac{3}{1 - a}$

2) $\frac{b^2 + 3b}{b^3 + 9b} * (\frac{b - 3}{b + 3} + \frac{b + 3}{b - 3}) = \frac{b(b + 3)}{b(b^2 + 9)} * \frac{(b - 3)^2 + (b + 3)^2}{(b - 3)(b + 3)} = \frac{b + 3}{b^2 + 9} * \frac{b^2 - 6b + 9 + b^2 + 6b + 9}{(b - 3)(b + 3)} = \frac{1}{b^2 + 9} * \frac{2b^2 + 18}{b - 3} = \frac{1}{b^2 + 9} * \frac{2(b^2 + 9)}{b - 3} = \frac{1}{1} * \frac{2}{b - 3} = \frac{2}{b - 3}$

3) $(\frac{3c + 1}{3c - 1} - \frac{3c - 1}{3c + 1}) : \frac{2c}{6c + 2} = \frac{(3c + 1)^2 - (3c - 1)^2}{(3c - 1)(3c + 1)} : \frac{2c}{2(3c + 1)} = \frac{9c^2 + 6c + 1 - (9c^2 - 6c + 1)}{(3c - 1)(3c + 1)} * \frac{2(3c + 1)}{2c} = \frac{12c}{3c - 1} * \frac{2}{2c} = \frac{6}{3c - 1} * \frac{2}{1} = \frac{12}{3c - 1}$

4) $(\frac{1}{a^2 - 4ab + 4b^2} - \frac{1}{4b^2 - a^2}) : \frac{2a}{a^2 - 4b^2} = (\frac{1}{(a - 2b)^2} - \frac{1}{(2b - a)(2b + a)}) : \frac{2a}{(a - 2b)(a + 2b)} = (\frac{1}{(2b - a)^2} - \frac{1}{(2b - a)(2b + a)}) : \frac{2a}{(a - 2b)(a + 2b)} = \frac{2b + a - (2b - a)}{(2b - a)^2(2b + a)} * \frac{(a - 2b)(a + 2b)}{2a} = \frac{2b + a - 2b + a}{(a - 2b)^2(a + 2b)} * \frac{(a - 2b)(a + 2b)}{2a} = \frac{2a}{(a - 2b)^2(a + 2b)} * \frac{(a - 2b)(a + 2b)}{2a} = \frac{1}{a - 2b} * \frac{1}{1} = \frac{1}{a - 2b}$

5) $(\frac{a - 8}{a^2 - 10a + 25} - \frac{a}{a^2 - 25}) : \frac{a - 20}{(a - 5)^2} = (\frac{a - 8}{(a - 5)^2} - \frac{a}{(a - 5)(a + 5)}) : \frac{a - 20}{(a - 5)^2} = \frac{(a - 8)(a + 5) - a(a - 5)}{(a - 5)^2(a + 5)} * \frac{(a - 5)^2}{a - 20} = \frac{a^2 - 8a + 5a - 40 - a^2 + 5a}{a + 5} * \frac{1}{a - 20} = \frac{2a - 40}{a + 5} * \frac{1}{a - 20} = \frac{2(a - 20)}{a + 5} * \frac{1}{a - 20} = \frac{2}{a + 5}$

6) $(\frac{2x + 1}{x^2 + 6x + 9} - \frac{x - 2}{x^2 + 3x}) : \frac{x^2 + 6}{x^3 - 9x} = (\frac{2x + 1}{(x + 3)^2} - \frac{x - 2}{x(x + 3)}) : \frac{x^2 + 6}{x(x^2 - 9)} = \frac{x(2x + 1) - (x - 2)(x + 3)}{x(x + 3)^2} : \frac{x^2 + 6}{x(x - 3)(x + 3)} = \frac{2x^2 + x - (x^2 - 2x + 3x - 6)}{x(x + 3)^2} * \frac{x(x - 3)(x + 3)}{x^2 + 6} = \frac{2x^2 + x - x^2 + 2x - 3x + 6}{x(x + 3)} * \frac{x(x - 3)}{x^2 + 6} = \frac{x^2 + 6}{x(x + 3)} * \frac{x(x - 3)}{x^2 + 6} = \frac{1}{x + 3} * \frac{x - 3}{1} = \frac{x - 3}{x + 3}$

179. Выполните действия:
1) $\frac{b + 4}{b^2 - 6b + 9} : \frac{b^2 - 16}{2b - 6} - \frac{2}{b - 4}$;
2) $(\frac{m - 1}{m + 1} - \frac{m + 1}{m - 1}) : \frac{4m}{m^2 - 1}$;
3) $\frac{2x}{x^2 - y^2} : (\frac{1}{x^2 + 2xy + y^2} - \frac{1}{y^2 - x^2})$;
4) $(\frac{2a - 3}{a^2 - 4a + 4} - \frac{a - 1}{a^2 - 2a}) : \frac{a^2 - 2}{a^3 - 4a}$.

Решение:

1) $\frac{b + 4}{b^2 - 6b + 9} : \frac{b^2 - 16}{2b - 6} - \frac{2}{b - 4} = \frac{b + 4}{(b - 3)^2} : \frac{(b - 4)(b + 4)}{2(b - 3)} - \frac{2}{b - 4} = \frac{b + 4}{(b - 3)^2} * \frac{2(b - 3)}{(b - 4)(b + 4)} - \frac{2}{b - 4} = \frac{1}{b - 3} * \frac{2}{b - 4} - \frac{2}{b - 4} = \frac{2}{b - 4}(\frac{1}{b - 3} - 1) = \frac{2}{b - 4} * \frac{1 - (b - 3)}{b - 3} = \frac{2}{b - 4} * \frac{1 - b + 3}{b - 3} = \frac{2}{b - 4} * \frac{4 - b}{b - 3} = \frac{2}{b - 4} * \frac{b - 4}{3 - b} = \frac{2}{1} * \frac{1}{3 - b} = \frac{2}{3 - b}$

2) $(\frac{m - 1}{m + 1} - \frac{m + 1}{m - 1}) : \frac{4m}{m^2 - 1} = \frac{(m - 1)^2 - (m + 1)^2}{(m - 1)(m + 1)} : \frac{4m}{m^2 - 1} = \frac{m^2 - 2m + 1 - (m^2 + 2m + 1)}{m^2 - 1} * \frac{m^2 - 1}{4m} = \frac{m^2 - 2m + 1 - m^2 - 2m - 1}{1} * \frac{1}{4m} = \frac{-4m}{4m} = -1$

3) $\frac{2x}{x^2 - y^2} : (\frac{1}{x^2 + 2xy + y^2} - \frac{1}{y^2 - x^2}) = \frac{2x}{x^2 - y^2} : (\frac{1}{(x + y)^2} + \frac{1}{x^2 - y^2}) = \frac{2x}{x^2 - y^2} : (\frac{1}{(x + y)^2} + \frac{1}{(x - y)(x + y)}) = \frac{2x}{x^2 - y^2} : \frac{x - y + x + y}{(x + y)^2(x - y)} = \frac{2x}{x^2 - y^2} : \frac{2x}{(x + y)^2(x - y)} = \frac{2x}{(x - y)(x + y)} * \frac{(x + y)^2(x - y)}{2x} = \frac{1}{1} * \frac{x + y}{1} = x + y$

4) $(\frac{2a - 3}{a^2 - 4a + 4} - \frac{a - 1}{a^2 - 2a}) : \frac{a^2 - 2}{a^3 - 4a} = (\frac{2a - 3}{(a - 2)^2} - \frac{a - 1}{a(a - 2)}) : \frac{a^2 - 2}{a(a^2 - 4)} = \frac{a(2a - 3) - (a - 1)(a - 2)}{a(a - 2)^2} : \frac{a^2 - 2}{a(a - 2)(a + 2)} = \frac{2a^2 - 3a - (a^2 - a - 2a + 2)}{a(a - 2)^2} * \frac{a(a - 2)(a + 2)}{a^2 - 2} = \frac{2a^2 - 3a - a^2 + a + 2a - 2}{a - 2} * \frac{a + 2}{a^2 - 2} = \frac{a^2 - 2}{a - 2} * \frac{a + 2}{a^2 - 2} = \frac{1}{a - 2} * \frac{a + 2}{1} = \frac{a + 2}{a - 2}$

180. Упростите выражение:
1) $(\frac{15}{x - 7} - x - 7) * \frac{7 - x}{x^2 - 16x + 64}$;
2) $(a - \frac{5a - 16}{a - 3}) : (2a - \frac{2a}{a - 3})$;
3) $(\frac{1}{a} + \frac{2}{b} + \frac{a}{b^2}) * \frac{ab}{a^2 - b^2} + \frac{2}{b - a}$;
4) $(\frac{a}{a - 1} - \frac{a}{a + 1} - \frac{a^2 + 1}{1 - a^2}) : \frac{a^2 + a}{(a - 1)^2}$;
5) $(\frac{x + 2y}{x - 2y} - \frac{x - 2y}{x + 2y} - \frac{16y^2}{x^2 - 4y^2}) : \frac{4y}{x + 2y}$;
6) $(\frac{3a - 8}{a^2 - 2a + 4} + \frac{1}{a + 2} - \frac{4a - 28}{a^3 + 8}) * \frac{a^2 - 4}{4}$.

Решение:

1) $(\frac{15}{x - 7} - x - 7) * \frac{7 - x}{x^2 - 16x + 64} = \frac{15 - x(x - 7) - 7(x - 7))}{x - 7} * \frac{7 - x}{(x - 8)^2} = \frac{15 - x^2 + 7x - 7x + 49}{x - 7} * \frac{7 - x}{(x - 8)^2} = \frac{64 - x^2}{x - 7} * \frac{7 - x}{(x - 8)^2} = \frac{x^2 - 64}{7 - x} * \frac{7 - x}{(x - 8)^2} = \frac{(x - 8)(x + 8)}{1} * \frac{1}{(x - 8)^2} = \frac{x + 8}{x - 8}$

2) $(a - \frac{5a - 16}{a - 3}) : (2a - \frac{2a}{a - 3}) = \frac{a(a - 3) - (5a - 16)}{a - 3} : \frac{2a(a - 3) - 2a}{a - 3} = \frac{a^2 - 3a - 5a + 16}{a - 3} : \frac{2a^2 - 6a - 2a}{a - 3} = \frac{a^2 - 8a + 16}{a - 3} : \frac{2a^2 - 8a}{a - 3} = \frac{(a - 4)^2}{a - 3} : \frac{2a(a - 4)}{a - 3} = \frac{(a - 4)^2}{a - 3} * \frac{a - 3}{2a(a - 4)} = \frac{a - 4}{1} * \frac{1}{2a} = \frac{a - 4}{2a}$

3) $(\frac{1}{a} + \frac{2}{b} + \frac{a}{b^2}) * \frac{ab}{a^2 - b^2} + \frac{2}{b - a} = \frac{b^2 + 2ab + a^2}{ab^2} * \frac{ab}{(a - b)(a + b)} + \frac{2}{b - a} = \frac{(a + b)^2}{ab^2} * \frac{ab}{(a - b)(a + b)} + \frac{2}{b - a} = \frac{a + b}{b} * \frac{1}{a - b} + \frac{2}{b - a} = \frac{a + b}{b(a - b)} - \frac{2}{a - b} = \frac{a + b - 2b}{b(a - b)} = \frac{a - b}{b(a - b)} = \frac{1}{b}$

4) $(\frac{a}{a - 1} - \frac{a}{a + 1} - \frac{a^2 + 1}{1 - a^2}) : \frac{a^2 + a}{(a - 1)^2} = (\frac{a}{a - 1} - \frac{a}{a + 1} + \frac{a^2 + 1}{a^2 - 1}) : \frac{a^2 + a}{(a - 1)^2} = (\frac{a}{a - 1} - \frac{a}{a + 1} + \frac{a^2 + 1}{(a - 1)(a + 1)}) : \frac{a^2 + a}{(a - 1)^2} = \frac{a(a + 1) - a(a - 1) + a^2 + 1}{(a - 1)(a + 1)} : \frac{a(a + 1)}{(a - 1)^2} = \frac{a^2 + a - a^2 + a + a^2 + 1}{(a - 1)(a + 1)} : \frac{a(a + 1)}{(a - 1)^2} = \frac{a^2 + 2a + 1}{(a - 1)(a + 1)} : \frac{a(a + 1)}{(a - 1)^2} = \frac{(a + 1)^2}{(a - 1)(a + 1)} * \frac{(a - 1)^2}{a(a + 1)} = \frac{1}{1} * \frac{a - 1}{a} = \frac{a - 1}{a}$

5) $(\frac{x + 2y}{x - 2y} - \frac{x - 2y}{x + 2y} - \frac{16y^2}{x^2 - 4y^2}) : \frac{4y}{x + 2y} = (\frac{x + 2y}{x - 2y} - \frac{x - 2y}{x + 2y} - \frac{16y^2}{(x - 2y)(x + 2y)}) : \frac{4y}{x + 2y} = \frac{(x + 2y)^2 - (x - 2y)^2 - 16y^2}{(x - 2y)(x + 2y)} * \frac{x + 2y}{4y} = \frac{(x + 2y - (x - 2y))(x + 2y + x - 2y) - 16y^2}{x - 2y} * \frac{1}{4y} = \frac{2x(x + 2y - x + 2y) - 16y^2}{x - 2y} * \frac{1}{4y} = \frac{2x * 4y - 16y^2}{x - 2y} * \frac{1}{4y} = \frac{4y(2x - 4y)}{x - 2y} * \frac{1}{4y} = \frac{2x - 4y}{x - 2y} = \frac{2(x - 2y)}{x - 2y} = 2$

6) $(\frac{3a - 8}{a^2 - 2a + 4} + \frac{1}{a + 2} - \frac{4a - 28}{a^3 + 8}) * \frac{a^2 - 4}{4} = (\frac{3a - 8}{a^2 - 2a + 4} + \frac{1}{a + 2} - \frac{4a - 28}{(a + 2)(a^2 - 2a + 4)}) * \frac{a^2 - 4}{4} = \frac{(3a - 8)(a + 2) + a^2 - 2a + 4 - (4a - 28)}{(a + 2)(a^2 - 2a + 4)} * \frac{a^2 - 4}{4} = \frac{3a^2 - 8a + 6a - 16 + a^2 - 2a + 4 - 4a + 28}{(a + 2)(a^2 - 2a + 4)} * \frac{a^2 - 4}{4} = \frac{4a^2 - 8a + 16}{(a + 2)(a^2 - 2a + 4)} * \frac{(a - 2)(a + 2)}{4} = \frac{4(a^2 - 2a + 4)}{a^2 - 2a + 4} * \frac{a - 2}{4} = \frac{1}{1} * \frac{a - 2}{1} = a - 2$