Ответы к странице 17

45. Сократите дробь:
1) $\frac{2m^2 - 72n^2}{(4m + 24n)^2}$;
2) $\frac{a^3 - 8}{ab - a - 2b + 2}$;
3) $\frac{a^3 + 2a^2b + ab^2}{a^3 - ab^2}$.

Решение:

1) $\frac{2m^2 - 72n^2}{(4m + 24n)^2} = \frac{2(m^2 - 36n^2)}{16(m + 6n)^2} = \frac{2(m - 6n)(m + 6n)}{16(m + 6n)^2} = \frac{m - 6n}{8(m + 6n)}$

2) $\frac{a^3 - 8}{ab - a - 2b + 2} = \frac{(a - 2)(a^2 + 2a + 4)}{(ab - a) - (2b - 2)} = \frac{(a - 2)(a^2 + 2a + 4)}{a(b - 1) - 2(b - 1)} = \frac{(a - 2)(a^2 + 2a + 4)}{(b - 1)(a - 2)} = \frac{a^2 + 2a + 4}{b - 1}$

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

46. Найдите значение дроби, предварительно сократив ее:
1) $\frac{15a^2 + 10ab}{3ab + 2b^2}$, если a = −2, b = 0,4;
2) $\frac{9b^2 - 4c^2}{12b^2c - 8bc^2}$, если $b = \frac{1}{3}$, c = −6;
3) $\frac{36x^2 - 12xy + y^2}{y^2 - 36x^2}$, если x = 1,2, y = −3;
4) $\frac{a^8 - a^6}{a^9 + a^8}$, если a = −0,1.

Решение:

1) $\frac{15a^2 + 10ab}{3ab + 2b^2} = \frac{5a(3a + 2b)}{b(3a + 2b)} = \frac{5a}{b}$
при a = −2, b = 0,4:
$\frac{5 * (-2)}{0,4} = \frac{-10}{\frac{4}{10}} = -10 * \frac{10}{4} = -5 * 5 = -25$

2) $\frac{9b^2 - 4c^2}{12b^2c - 8bc^2} = \frac{(3b - 2c)(3b + 2c)}{4bc(3b - 2c)} = \frac{3b + 2c}{4bc}$
при $b = \frac{1}{3}$, c = −6:
$\frac{3 * \frac{1}{3} + 2 * (-6)}{4 * \frac{1}{3} * (-6)} = \frac{1 - 12}{4 * (-2)} = \frac{-11}{-8} = \frac{11}{8} = 1\frac{3}{8}$

3) $\frac{36x^2 - 12xy + y^2}{y^2 - 36x^2} = \frac{(6x - y)^2}{(y - 6x)(y + 6x)} = \frac{(y - 6x)^2}{(y - 6x)(y + 6x)} = \frac{y - 6x}{y + 6x}$
при x = 1,2, y = −3:
$\frac{-3 - 6 * 1,2}{-3 + 6 * 1,2} = \frac{-3 - 7,2}{-3 + 7,2} = \frac{-10,2}{4,2} = -\frac{102}{42} = -\frac{17}{7} = -2\frac{3}{7}$

4) $\frac{a^8 - a^6}{a^9 + a^8} = \frac{a^6(a^2 - 1)}{a^8(a + 1)} = \frac{a^6(a - 1)(a + 1)}{a^8(a + 1)} = \frac{a - 1}{a^2}$
при a = −0,1:
$\frac{-0,1 - 1}{(-0,1)^2} = \frac{-1,1}{0,01} = -110$

47. Найдите значение выражения:
1) $\frac{16x^2 - 4y^2}{6x - 3y}$ при x = 2,5, y = −2;
2) $\frac{49c^2 - 9}{49c^2 + 42c + 9}$ при c = −4.

Решение:

1) $\frac{16x^2 - 4y^2}{6x - 3y} = \frac{(4x - 2y)(4x + 2y)}{3(2x - y)} = \frac{2 * 2(2x - y)(2x + y)}{3(2x - y)} = \frac{4(2x + y)}{3}$
при x = 2,5, y = −2:
$\frac{4(2 * 2,5 - 2)}{3} = \frac{4(5 - 2)}{3} = \frac{4 * 3}{3} = 4$

2) $\frac{49c^2 - 9}{49c^2 + 42c + 9} = \frac{(7c - 3)(7c + 3)}{(7c + 3)^2} = \frac{7c - 3}{7c + 3}$
при c = −4:
$\frac{7 * (-4) - 3}{7 * (-4) + 3} = \frac{-28 - 3}{-28 + 3} = \frac{-31}{-25} = 1\frac{6}{25}$

48. Приведите к общему знаменателю дроби:
1) $\frac{2p}{5p - 15}$ и $\frac{1}{p^3 - 27}$;
2) $\frac{3a + 1}{9a^2 - 6a + 1}$ и $\frac{a - 2}{9a^2 - 1}$;
3) $\frac{a}{a^2 - 7a}$ и $\frac{a + 3}{a^2 - 14a + 49}$;
4) $\frac{2x}{x^2 - 1}, \frac{3x}{x^2 - 2x + 1}$ и $\frac{4}{x^2 + 2x + 1}$;
5) $\frac{a^2}{a^2 - ab - ac + bc}, \frac{b}{2a - 2b}$ и $\frac{ab}{4a - 4c}$.

Решение:

1) $\frac{2p}{5p - 15} = \frac{2p}{5(p - 3)} = \frac{2p(p^2 + 3p + 9)}{5(p - 3)(p^2 + 3p + 9)} = \frac{2p(p^2 + 3p + 9)}{5(p^3 - 27)}$
$\frac{1}{p^3 - 27} = \frac{1}{(p - 3)(p^2 + 3p + 9)} = \frac{5}{5(p - 3)(p^2 + 3p + 9)} = \frac{5}{5(p^3 - 27)}$

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

3) $\frac{a}{a^2 - 7a} = \frac{a}{a(a - 7)} = \frac{a - 7}{(a - 7)(a - 7)} = \frac{a - 7}{(a - 7)^2}$
$\frac{a + 3}{a^2 - 14a + 49} = \frac{a + 3}{(a - 7)^2} = \frac{a(a + 3)}{a(a - 7)^2} = \frac{a + 3}{(a - 7)^2}$

4) $\frac{2x}{x^2 - 1} = \frac{2x}{(x - 1)(x + 1)} = \frac{2x(x - 1)(x + 1)}{(x - 1)(x + 1)(x - 1)(x + 1)} = \frac{2x(x^2 - 1)}{(x - 1)^2(x + 1)^2}$
$\frac{3x}{x^2 - 2x + 1} = \frac{3x}{(x - 1)^2} = \frac{3x(x + 1)^2}{(x - 1)^2(x + 1)^2}$
$\frac{4}{x^2 + 2x + 1} = \frac{4}{(x + 1)^2} = \frac{4(x - 1)^2}{(x - 1)^2(x + 1)^2}$

5) $\frac{a^2}{a^2 - ab - ac + bc} = \frac{a^2}{(a^2 - ab) - (ac - bc)} = \frac{a^2}{a(a - b) - c(a - b)} = \frac{a^2}{(a - b)(a - c)} = \frac{4a^2}{4(a - b)(a - c)}$
$\frac{b}{2a - 2b} = \frac{b}{2(a - b)} = \frac{2 * b(a - c)}{2 * 2(a - b)(a - c)} = \frac{2b(a - c)}{4(a - b)(a - c)}$
$\frac{ab}{4a - 4c} = \frac{ab}{4(a - c)} = \frac{ab(a - b)}{4(a - b)(a - c)}$

49. Запишите в виде дробей с одинаковыми знаменателями:
1) $\frac{3a}{3a - 2}, \frac{a}{9a + 6}$ и $\frac{a^2}{9a^2b - 4b}$;
2) $\frac{1}{a - 5b}, \frac{1}{a^2 + 7ac}$ и $\frac{1}{a^2 + 7ac - 5ab - 35bc}$.

Решение:

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

2) $\frac{1}{a - 5b} = \frac{a(a + 7c)}{a(a - 5b)(a + 7c)}$
$\frac{1}{a^2 + 7ac} = \frac{1}{a(a + 7c)} = \frac{a - 5b}{a(a - 5b)(a + 7c)}$
$\frac{1}{a^2 + 7ac - 5ab - 35bc} = \frac{1}{(a^2 + 7ac) - (5ab + 35bc)} = \frac{1}{a(a + 7c) - 5b(a + 7c)} = \frac{a}{a(a + 7c)(a - 5b)}$

50. Найдите значение выражения
$\frac{2xy - y^2}{3xy + x^2}$, если $\frac{x}{y} = 2$.

Решение:

$\frac{2xy - y^2}{3xy + x^2} = \frac{y(2x - y)}{x(3y + x)} = \frac{y}{x} * \frac{2x - y}{3y + x}$
при $\frac{x}{y} = 2$
$\frac{y}{x} = \frac{1}{2}$
x = 2y:
$\frac{y}{x} * \frac{2x - y}{3y + x} = \frac{1}{2} * \frac{2 * 2y - y}{3y + 2y} = \frac{1}{2} * \frac{4y - y}{5y} = \frac{3y}{10y} = 0,3$

51. Найдите значение выражения
$\frac{4a^2 - ab}{ab + 14b^2}$, если $\frac{a}{b} = 5$.

Решение:

$\frac{4a^2 - ab}{ab + 14b^2} = \frac{a(4a - b)}{b(a + 14b)} = \frac{a}{b} * \frac{4a - b}{a + 14b}$
при $\frac{a}{b} = 5$
a = 5b
$5 * \frac{4 * 5b - b}{5b + 14b} = 5 * \frac{20b - b}{19b} = 5 * \frac{19b}{19b} = 5 * 1 = 5$

52. Известно, что 2a − 6b = 1. Найдите значение выражения:
1) $\frac{8}{a - 3b}$;
2) $\frac{a^2 - 9b^2}{0,5a + 1,5b}$.

Решение:

1) 2a − 6b = 1
2(a − 3b) = 1
$a - 3b = \frac{1}{2}$, тогда:
$\frac{8}{a - 3b} = \frac{8}{\frac{1}{2}} = 8 * 2 = 16$

2) $\frac{a^2 - 9b^2}{0,5a + 1,5b} = \frac{(a - 3b)(a + 3b)}{0,5(a + 3b)} = \frac{a - 3b}{\frac{1}{2}} = 2(a - 3b) = 2a - 6b = 1$