Ответы к странице 16

39. Приведите дробь:
1) $\frac{a}{a + 2}$ к знаменателю 4a + 8;
2) $\frac{m}{m - 3n}$ к знаменателю $m^2 - 9n^2$;
3) $\frac{x}{2x - y}$ к знаменателю 7y − 14x;
4) $\frac{5b}{2a + 3b}$ к знаменателю $4a^2 + 12ab + 9b^2$;
5) $\frac{x + 1}{x^2 + x + 1}$ к знаменателю $x^3 - 1$.

Решение:

1) $\frac{4a + 8}{a + 2} = \frac{4(a + 2)}{a + 2} = 4$, тогда:
$\frac{a}{a + 2} = \frac{4 * a}{4 * (a + 2)} = \frac{4a}{4a + 8}$

2) $\frac{m^2 - 9n^2}{m - 3n} = \frac{(m - 3n)(m + 3n)}{m - 3n} = m + 3n$, тогда:
$\frac{m}{m - 3n} = \frac{(m + 3n) * m}{(m + 3) * (m - 3n)} = \frac{m^2 + 3mn}{m^2 - 9n^2}$

3) $\frac{7y - 14x}{2x - y} = \frac{7(y - 2x)}{2x - y} = -\frac{7(2x - y)}{2x - y} = -7$, тогда:
$\frac{x}{2x - y} = \frac{-7 * x}{-7 * (2x - y)} = \frac{-7x}{-14x + 7y} = -\frac{7x}{7y - 14x}$

4) $\frac{4a^2 + 12ab + 9b^2}{2a + 3b} = \frac{(2a + 3b)^2}{2a + 3b} = 2a + 3b$, тогда:
$\frac{5b}{2a + 3b} = \frac{(2a + 3b) * 5b}{(2a + 3b) * (2a + 3b)} = \frac{10ab + 15b^2}{(2a + 3b)^2} = \frac{10ab + 15b^2}{4a^2 + 12ab + 9b^2}$

5) $\frac{x^3 - 1}{x^2 + x + 1} = \frac{(x - 1)(x^2 + x + 1)}{x^2 + x + 1} = x - 1$, тогда:
$\frac{x + 1}{x^2 + x + 1} = \frac{(x - 1) * (x + 1)}{(x - 1) * (x^2 + x + 1)} = \frac{x^2 - 1}{x^3 - 1}$

40. Представьте выражение x − 5y в виде дроби со знаменателем:
1) 2;
2) x;
3) $4y^3$;
4) $x^2 - 25y^2$.

Решение:

1) $x - 5y = \frac{x - 5y}{1} = \frac{2 * (x - 5y)}{2 * 1} = \frac{2x - 10y}{2}$

2) $x - 5y = \frac{x - 5y}{1} = \frac{x * (x - 5y)}{x * 1} = \frac{x^2 - 5xy}{x}$

3) $x - 5y = \frac{x - 5y}{1} = \frac{4y^3 * (x - 5y)}{4y^3 * 1} = \frac{4xy^3 - 20y^4}{4y^3}$

4) $x - 5y = \frac{(x^2 - 25y^2) * (x - 5y)}{(x^2 - 25y^2) * 1} = \frac{(x - 5y)(x + 5y)(x - 5y)}{x^2 - 25y^2} = \frac{(x - 5y)^2(x + 5y)}{x^2 - 25y^2}$

41. Приведите дробь $\frac{6}{b - 4}$ к знаменателю:
1) 5b − 20;
2) 12 − 3b;
3) $b^2 - 4b$;
4) $b^2 - 16$.

Решение:

1) $\frac{5b - 20}{b - 4} = \frac{5(b - 4)}{b - 4} = 5$, тогда:
$\frac{6}{b - 4} = \frac{5 * 6}{5 * (b - 4)} = \frac{30}{5b - 20}$

2) $\frac{12 - 3b}{b - 4} = \frac{3(4 - b)}{b - 4} = -\frac{3(b - 4)}{b - 4} = -3$, тогда:
$\frac{6}{b - 4} = \frac{-3 * 6}{-3 * (b - 4)} = \frac{-18}{-3b + 12} = -\frac{18}{12 - 3b}$

3) $\frac{b^2 - 4b}{b - 4} = \frac{b(b - 4)}{b - 4} = b$, тогда:
$\frac{6}{b - 4} = \frac{b * 6}{b * (b - 4)} = \frac{6b}{b^2 - 4b}$

4) $\frac{b^2 - 16}{b - 4} = \frac{(b - 4)(b + 4)}{b - 4} = b + 4$, тогда:
$\frac{6}{b - 4} = \frac{(b + 4) * 6}{(b + 4) * (b - 4)} = \frac{6b + 24}{b^2 - 16}$

42. Представьте данные дроби в виде дробей с одинаковыми знаменателями:
1\ $\frac{1}{8ab}$ и $\frac{1}{2a^3}$;
2) $\frac{3x}{7m^3n^3}$ и $\frac{4y}{3m^2n^4}$;
3) $\frac{a + b}{a - b}$ и $\frac{2}{a^2 - b^2}$;
4) $\frac{3d}{m - n}$ и $\frac{8p}{(m - n)^2}$;
5) $\frac{x}{2x + 1}$ и $\frac{x}{3x - 2}$;
6) $\frac{a - b}{3a + 3b}$ и $\frac{a}{a^2 - b^2}$;
7) $\frac{3a}{4a - 4}$ и $\frac{2a}{5 - 5a}$;
8) $\frac{7a}{b - 3}$ и $\frac{c}{9 - b^2}$.

Решение:

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

2) $\frac{3x}{7m^3n^3} = \frac{3x * 3n}{7m^3n^3 * 3n} = \frac{9nx}{21m^3n^4}$
$\frac{4y}{3m^2n^4} = \frac{4y * 7m}{3m^2n^4 * 7m} = \frac{28my}{21m^3n^4}$

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

4) $\frac{3d}{m - n} = \frac{3d * (m - n)}{(m - n) * (m - n)} = \frac{3d(m - n)}{(m - n)^2}$
$\frac{8p}{(m - n)^2}$

5) $\frac{x}{2x + 1} = \frac{x(3x - 2)}{(2x + 1)(3x - 2)} = \frac{3x^2 - 2x}{6x^2 + 3x - 4x - 2} = \frac{3x^2 - 2x}{6x^2 - x - 2}$
$\frac{x}{3x - 2} = \frac{x(2x + 1)}{(3x - 2)(2x + 1)} = \frac{2x^2 + x}{6x^2 + 3x - 4x - 2} = \frac{2x^2 + x}{6x^2 - x - 2}$

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

7) $\frac{3a}{4a - 4} = \frac{3a}{4(a - 1)} = \frac{5 * 3a}{5 * 4(a - 1)} = \frac{15a}{20(a - 1)}$
$\frac{2a}{5 - 5a} = \frac{2a}{5(1 - a)} = -\frac{4 * 2a}{4 * 5(a - 1)} = -\frac{8a}{20(a - 1)}$

8) $\frac{7a}{b - 3} = -\frac{7a}{3 - b} = -\frac{7a(3 + b)}{(3 - b)(3 + b)} = -\frac{7a(3 + b)}{9 - b^2}$
$\frac{c}{9 - b^2}$

43. Приведите к общему знаменателю дроби:
1) $\frac{4}{15x^2y^2}$ и $\frac{1}{10x^3y}$;
2) $\frac{c}{6a^4b^5}$ и $\frac{d}{9ab^2}$;
3) $\frac{x}{y - 5}$ и $\frac{z}{y^2 - 25}$;
4) $\frac{m + n}{m^2 - mn}$ и $\frac{2m - 3n}{m^2 - n^2}$;
5) $\frac{x + 1}{x^2 - xy}$ и $\frac{y - 1}{xy - y^2}$;
6) $\frac{6a}{a - 2b}$ и $\frac{3a}{a + b}$;
7) $\frac{1 + c^2}{c^2 - 16}$ и $\frac{c}{4 - c}$;
8) $\frac{2m + 9}{m^2 + 5m + 25}$ и $\frac{m}{m - 5}$.

Решение:

1) $\frac{4}{15x^2y^2} = \frac{4 * 2x}{15x^2y^2 * 2x} = \frac{8x}{30x^3y^2}$
$\frac{1}{10x^3y} = \frac{1 * 3y}{10x^3y * 3y} = \frac{3y}{30x^3y^2}$

2) $\frac{c}{6a^4b^5} = \frac{c * 3}{6a^4b^5 * 3} = \frac{3c}{18a^4b^5}$
$\frac{d}{9ab^2} = \frac{d * 2a^3b^3}{9ab^2 * 2a^3b^3} = \frac{2a^3b^3d}{18a^4b^5}$

3) $\frac{x}{y - 5} = \frac{x(y + 5)}{(y - 5)(y + 5)} = \frac{x(y + 5)}{y^2 - 25}$
$\frac{z}{y^2 - 25}$

4) $\frac{m + n}{m^2 - mn} = \frac{m + n}{m(m - n)} = \frac{(m + n)(m + n)}{m(m - n)(m + n)} = \frac{(m + n)^2}{m(m^2 - n^2)}$
$\frac{2m - 3n}{m^2 - n^2} = \frac{2m - 3n}{(m - n)(m + n)} = \frac{m(2m - 3n)}{m(m^2 - n^2)}$

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

6) $\frac{6a}{a - 2b} = \frac{6a(a + b)}{(a - 2b)(a + b)}$
$\frac{3a}{a + b} = \frac{3a(a - 2b)}{(a - 2b)(a + b)}$

7) $\frac{1 + c^2}{c^2 - 16} = \frac{1 + c^2}{(c - 4)(c + 4)} = \frac{1 + c^2}{c^2 - 16}$
$\frac{c}{4 - c} = -\frac{c}{c - 4} = -\frac{c(c + 4)}{(c - 4)(c + 4)} = -\frac{c(c + 4)}{c^2 - 16}$

8) $\frac{2m + 9}{m^2 + 5m + 25} = \frac{(m - 5)(2m + 9)}{(m - 5)(m^2 + 5m + 25)} = \frac{(m - 5)(2m + 9)}{m^3 - 125}$
$\frac{m}{m - 5} = \frac{m(m^2 + 5m + 25)}{(m - 5)(m^2 + 5m + 25)} = \frac{m(m^2 + 5m + 25)}{m^3 - 125}$

44. Сократите:
1) $\frac{(3a + 3b)^2}{a + b}$;
2) $\frac{(6x - 18y)^2}{x^2 - 9y^2}$;
3) $\frac{xy + x - 5y - 5}{4y + 4}$;
4) $\frac{a^2 - ab + 2b - 2a}{a^2 - 4a + 4}$.

Решение:

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

2) $\frac{(6x - 18y)^2}{x^2 - 9y^2} = \frac{36(x - 3y)^2}{(x - 3y)(x + 3y)} = \frac{36(x - 3y)}{x + 3y}$

3) $\frac{xy + x - 5y - 5}{4y + 4} = \frac{(xy + x) - (5y + 5)}{4(y + 1)} = \frac{(x(y + 1) - 5(y + 1)}{4(y + 1)} = \frac{(y + 1)(x - 5)}{4(y + 1)} = \frac{x - 5}{4}$

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