Ответы к странице 43

§6. Тождественные преобразования рациональных выражений

Упражнения

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

Решение:

1) $(\frac{a}{3} + \frac{a}{4}) * \frac{6}{a^2} = \frac{4a + 3a}{12} * \frac{6}{a^2} = \frac{7a}{12} * \frac{6}{a^2} = \frac{7}{2} * \frac{1}{a} = \frac{7}{2a}$

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

3) $(1 + \frac{a}{b}) : (1 - \frac{a}{b}) = \frac{b + a}{b} : \frac{b - a}{b} = \frac{b + a}{b} * \frac{b}{b - a} = \frac{b + a}{1} * \frac{1}{b - a} = \frac{b + a}{b - a}$

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

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

6) $(\frac{5}{m - n} - \frac{4}{m + n}) : \frac{m + 9n}{m + n} = \frac{5(m + n) - 4(m - n)}{(m - n)(m + n)} : \frac{m + 9n}{m + n} = \frac{5m + 5n - 4m + 4n}{(m - n)(m + n)} : \frac{m + 9n}{m + n} = \frac{m + 9n}{(m - n)(m + n)} * \frac{m + n}{m + 9n} = \frac{1}{m - n} * \frac{1}{1} = \frac{1}{m - n}$

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

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

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

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