Ответы к странице 28

108. Выполните действия:
1) $\frac{3}{x + 3} + \frac{x + 4}{x^2 - 9}$;
2) $\frac{a^2}{a^2 - 64} - \frac{a}{a - 8}$;
3) $\frac{6b}{9b^2 - 4} - \frac{1}{3b - 2}$;
4) $\frac{3a + b}{a^2 - b^2} + \frac{1}{a + b}$;
5) $\frac{m}{m + 5} - \frac{m^2}{m^2 + 10m + 25}$;
6) $\frac{b}{a + b} - \frac{b^2}{a^2 + b^2 + 2ab}$.

Решение:

1) $\frac{3}{x + 3} + \frac{x + 4}{x^2 - 9} = \frac{3}{x + 3} + \frac{x + 4}{(x - 3)(x + 3)} = \frac{3(x - 3) + x + 4}{(x - 3)(x + 3)} = \frac{3x - 9 + x + 4}{(x - 3)(x + 3)} = \frac{4x - 5}{x^2 - 9}$

2) $\frac{a^2}{a^2 - 64} - \frac{a}{a - 8} = \frac{a^2}{(a - 8)(a + 8)} - \frac{a}{a - 8} = \frac{a^2 - a(a + 8)}{(a - 8)(a + 8)} = \frac{a^2 - a^2 - 8a}{a^2 - 64} = -\frac{8a}{a^2 - 64} = \frac{8a}{64 - a^2}$

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

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

5) $\frac{m}{m + 5} - \frac{m^2}{m^2 + 10m + 25} = \frac{m}{m + 5} - \frac{m^2}{(m + 5)^2} = \frac{m(m + 5) - m^2}{(m + 5)^2} = \frac{m^2 + 5m - m^2}{(m + 5)^2} = \frac{5m}{(m + 5)^2}$

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

109. Упростите выражение:
1) $\frac{4x - y}{x^2 - y^2} + \frac{1}{x - y}$;
2) $\frac{y^2}{y^2 - 81} - \frac{y}{y + 9}$;
3) $\frac{10a}{25a^2 - 9} - \frac{1}{5a + 3}$;
4) $\frac{n}{n - 7} - \frac{n^2}{n^2 - 14n + 49}$.

Решение:

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

2) $\frac{y^2}{y^2 - 81} - \frac{y}{y + 9} = \frac{y^2}{(y - 9)(y + 9)} - \frac{y}{y + 9} = \frac{y^2 - y(y - 9)}{(y - 9)(y + 9)} = \frac{y^2 - y^2 + 9y}{y^2 - 81} = \frac{9y}{y^2 - 81}$

3) $\frac{10a}{25a^2 - 9} - \frac{1}{5a + 3} = \frac{10a}{(5a - 3)(5a + 3)} - \frac{1}{5a + 3} = \frac{10a - (5a - 3)}{(5a - 3)(5a + 3)} = \frac{10a - 5a + 3}{(5a - 3)(5a + 3)} = \frac{5a + 3}{(5a - 3)(5a + 3)} = \frac{1}{5a - 3}$

4) $\frac{n}{n - 7} - \frac{n^2}{n^2 - 14n + 49} = \frac{n}{n - 7} - \frac{n^2}{(n - 7)^2} = \frac{n(n - 7) - n^2}{(n - 7)^2} = \frac{n^2 - 7n - n^2}{(n - 7)^2} = \frac{-7n}{(n - 7)^2} = -\frac{7n}{(n - 7)^2}$

110. Представьте в виде дроби выражение:
1) $\frac{a}{b} + 1$;
2) $\frac{x}{y} - x$;
3) $\frac{m}{n} + \frac{n}{m} + 2$;
4) $\frac{9}{p^2} - \frac{4}{p} + 3$;
5) $2 - \frac{3b + 2a}{a}$;
6) $\frac{3b + 4}{b - 2} - 3$;
7) $6m - \frac{12m^2 + 1}{2m}$;
8) $\frac{20b^2 + 5}{2b - 1} - 10b$.

Решение:

1) $\frac{a}{b} + 1 = \frac{a}{b} + \frac{b}{b} = \frac{a + b}{b}$

2) $\frac{x}{y} - x = \frac{x}{y} - \frac{xy}{y} = \frac{x - xy}{y}$

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

4) $\frac{9}{p^2} - \frac{4}{p} + 3 = \frac{9 - 4p + 3p^2}{p^2}$

5) $2 - \frac{3b + 2a}{a} = \frac{2a - (3b + 2a)}{a} = \frac{2a - 3b - 2a}{a} = -\frac{3b}{a}$

6) $\frac{3b + 4}{b - 2} - 3 = \frac{3b + 4 - 3(b - 2)}{b - 2} = \frac{3b + 4 - 3b + 6}{b - 2} = \frac{10}{b - 2}$

7) $6m - \frac{12m^2 + 1}{2m} = \frac{6m * 2m - (12m^2 + 1)}{2m} = \frac{12m^2 - 12m^2 - 1}{2m} = \frac{-1}{2m} = -\frac{1}{2m}$

8) $\frac{20b^2 + 5}{2b - 1} - 10b = \frac{20b^2 + 5 - 10b(2b - 1)}{2b - 1} = \frac{20b^2 + 5 - 20b^2 + 10b}{2b - 1} = \frac{10b + 5}{2b - 1}$

111. Выполните действия:
1) $a - \frac{4}{a}$;
2) $\frac{1}{x} + x - 2$;
3) $\frac{m}{n^3} - \frac{1}{n} + m$;
4) $\frac{2k^2}{k - 5} - k$;
5) $3n - \frac{9n^2 - 2}{3n}$;
6) $5 - \frac{4y - 12}{y - 2}$.

Решение:

1) $a - \frac{4}{a} = \frac{a^2 - 4}{a}$

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

3) $\frac{m}{n^3} - \frac{1}{n} + m = \frac{m - n^2 + mn^3}{n^3}$

4) $\frac{2k^2}{k - 5} - k = \frac{2k^2 - k(k - 5)}{k - 5} = \frac{2k^2 - k^2 + 5k}{k - 5} = \frac{k^2 + 5k}{k - 5}$

5) $3n - \frac{9n^2 - 2}{3n} = \frac{3n * 3n - (9n^2 - 2)}{3n} = \frac{9n^2 - 9n^2 + 2}{3n} = \frac{2}{3n}$

6) $5 - \frac{4y - 12}{y - 2} = \frac{5(y - 2) - (4y - 12)}{y - 2} = \frac{5y - 10 - 4y + 12}{y - 2} = \frac{y + 2}{y - 2}$

112. Упростите выражение:
1) $\frac{a^2 + 1}{a^2 - 2a + 1} + \frac{a + 1}{a - 1}$;
2) $\frac{a^2 + b^2}{a^2 - b^2} - \frac{a - b}{a + b}$;
3) $\frac{c + 7}{c - 7} + \frac{28c}{49 - c^2}$;
4) $\frac{5a + 3}{2a^2 + 6a} + \frac{6 - 3a}{a^2 - 9}$;
5) $\frac{a}{a^2 - 4a + 4} - \frac{a + 4}{a^2 - 4}$;
6) $\frac{2p}{p - 5} - \frac{5}{p + 5} + \frac{2p^2}{25 - p^2}$;
7) $\frac{1}{y} - \frac{y + 8}{16 - y^2} - \frac{2}{y - 4}$;
8) $\frac{2b - 1}{4b + 2} + \frac{4b}{4b^2 - 1} + \frac{2b + 1}{3 - 6b}$.

Решение:

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

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

3) $\frac{c + 7}{c - 7} + \frac{28c}{49 - c^2} = \frac{c + 7}{c - 7} - \frac{28c}{c^2 - 49} = \frac{c + 7}{c - 7} - \frac{28c}{(c - 7)(c + 7)} = \frac{(c + 7)(c + 7) - 28c}{(c - 7)(c + 7)} = \frac{(c + 7)^2 - 28c}{(c - 7)(c + 7)} = \frac{c^2 + 14c + 49 - 28c}{(c - 7)(c + 7)} = \frac{c^2 - 14c + 49}{(c - 7)(c + 7)} = \frac{(c - 7)^2}{(c - 7)(c + 7)} = \frac{c - 7}{c + 7}$

4) $\frac{5a + 3}{2a^2 + 6a} + \frac{6 - 3a}{a^2 - 9} = \frac{5a + 3}{2a(a + 3)} + \frac{6 - 3a}{(a - 3)(a + 3)} = \frac{(5a + 3)(a - 3) + 2a(6 - 3a)}{2a(a - 3)(a + 3)} = \frac{5a^2 + 3a - 15a - 9 + 12a - 6a^2}{2a(a - 3)(a + 3)} = \frac{-a^2 - 9}{2a(a^2 - 9)}$

5) $\frac{a}{a^2 - 4a + 4} - \frac{a + 4}{a^2 - 4} = \frac{a}{(a - 2)^2} - \frac{a + 4}{(a - 2)(a + 2)} = \frac{a(a + 2) - (a + 4)(a - 2)}{(a - 2)^2(a + 2)} = \frac{a^2 + 2a - (a^2 + 4a - 2a - 8)}{(a - 2)^2(a + 2)} = \frac{a^2 + 2a - a^2 - 4a + 2a + 8}{(a - 2)^2(a + 2)} = \frac{8}{(a - 2)^2(a + 2)}$

6) $\frac{2p}{p - 5} - \frac{5}{p + 5} + \frac{2p^2}{25 - p^2} = \frac{2p}{p - 5} - \frac{5}{p + 5} - \frac{2p^2}{p^2 - 25} = \frac{2p}{p - 5} - \frac{5}{p + 5} - \frac{2p^2}{(p - 5)(p + 5)} = \frac{2p(p + 5) - 5(p - 5) - 2p^2}{(p - 5)(p + 5)} = \frac{2p^2 + 10p - 5p + 25 - 2p^2}{(p - 5)(p + 5)} = \frac{5p + 25}{(p - 5)(p + 5)} = \frac{5(p + 5)}{(p - 5)(p + 5)} = \frac{5}{p - 5}$

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

8) $\frac{2b - 1}{4b + 2} + \frac{4b}{4b^2 - 1} + \frac{2b + 1}{3 - 6b} = \frac{2b - 1}{4b + 2} + \frac{4b}{4b^2 - 1} - \frac{2b + 1}{6b - 3} = \frac{2b - 1}{2(2b + 1)} + \frac{4b}{(2b - 1)(2b + 1)} - \frac{2b + 1}{3(2b - 1)} = \frac{3(2b - 1)(2b - 1) + 6 * 4b - 2(2b + 1)(2b + 1)}{6(2b - 1)(2b + 1)} = \frac{3(2b - 1)^2 + 24b - 2(2b + 1)^2}{6(2b - 1)(2b + 1)} = \frac{3(4b^2 - 4b + 1) + 24b - 2(4b^2 + 4b + 1)}{6(2b - 1)(2b + 1)} = \frac{12b^2 - 12b + 3 + 24b - 8b^2 - 8b - 2}{6(2b - 1)(2b + 1)} = \frac{4b^2 + 4b + 1}{6(2b - 1)(2b + 1)} = \frac{(2b + 1)^2}{6(2b - 1)(2b + 1)} = \frac{2b + 1}{6(2b - 1)}$