Ответы к странице 39

156. Упростите выражение:
1) $\frac{3a^4b^3}{10c^5} * \frac{4b^4c^2}{27a^7} : \frac{5b^7}{9a^3c^3}$;
2) $\frac{3a^2}{2b^2c^2} : \frac{7c^8}{6b^3} : \frac{9ab}{14c^{12}}$;
3) $(\frac{5a^3}{b^4})^4 * \frac{b^{18}}{50a^{16}}$;
4) $(\frac{3x^7}{y^{10}})^4 : (\frac{3x^6}{y^8})^3$.

Решение:

1) $\frac{3a^4b^3}{10c^5} * \frac{4b^4c^2}{27a^7} : \frac{5b^7}{9a^3c^3} = \frac{3a^4b^3}{10c^5} * \frac{4b^4c^2}{27a^7} * \frac{9a^3c^3}{5b^7} = \frac{1}{5} * \frac{2}{1} * \frac{1}{5} = \frac{2}{25}$

2) $\frac{3a^2}{2b^2c^2} : \frac{7c^8}{6b^3} : \frac{9ab}{14c^{12}} = \frac{3a^2}{2b^2c^2} * \frac{6b^3}{7c^8} * \frac{14c^{12}}{9ab} = \frac{a}{1} * \frac{1}{1} * \frac{2c^{2}}{1} = 2ac^2$

3) $(\frac{5a^3}{b^4})^4 * \frac{b^{18}}{50a^{16}} = \frac{5^4a^{12}}{b^{16}} * \frac{b^{18}}{2 * 5^2a^{16}} = \frac{5^2}{1} * \frac{b^{2}}{2a^{4}} = \frac{25b^2}{2a^4}$

4) $(\frac{3x^7}{y^{10}})^4 : (\frac{3x^6}{y^8})^3 = \frac{3^4x^{28}}{y^{40}} : \frac{3^3x^{18}}{y^{24}} = \frac{3^4x^{28}}{y^{40}} * \frac{y^{24}}{3^3x^{18}} = \frac{3x^{10}}{y^{16}} * \frac{1}{1} = \frac{3x^{10}}{y^{16}}$

157. Замените переменную x таким выражением, чтобы получилось тождество:
1) $(\frac{4a^2}{b^3})^2 * x = \frac{6a}{b^2}$;
2) $(\frac{2b^4}{3c})^3 : x = \frac{b^6}{12}$.

Решение:

1) $(\frac{4a^2}{b^3})^2 * x = \frac{6a}{b^2}$
$x = \frac{6a}{b^2} : (\frac{4a^2}{b^3})^2 = \frac{6a}{b^2} : \frac{16a^4}{b^6} = \frac{6a}{b^2} * \frac{b^6}{16a^4} = \frac{3}{1} * \frac{b^4}{8a^3} = \frac{3b^4}{8a^3}$
Ответ: $(\frac{4a^2}{b^3})^2 * \frac{3b^4}{8a^3} = \frac{6a}{b^2}$

2) $(\frac{2b^4}{3c})^3 : x = \frac{b^6}{12}$
$x = (\frac{2b^4}{3c})^3 : \frac{b^6}{12} = \frac{8b^{12}}{27c^3} : \frac{b^6}{12} = \frac{8b^{12}}{27c^3} * \frac{12}{b^6} = \frac{8b^{6}}{9c^3} * \frac{4}{1} = \frac{32b^{6}}{9c^3}$
Ответ: $(\frac{2b^4}{3c})^3 : \frac{32b^{6}}{9c^3} = \frac{b^6}{12}$

158. Выполните умножение и деление дробей:
1) $\frac{4 - a}{8a^3} * \frac{12a^5}{a^2 - 16}$;
2) $\frac{4c - d}{c^2 + cd} * \frac{2c^2 - 2d^2}{4c^2 - cd}$;
3) $\frac{b^2 - 6b + 9}{b^2 - 3b + 9} * \frac{b^3 + 27}{5b - 15}$;
4) $\frac{a^3 - 16a}{3a^2b} * \frac{12ab^2}{4a + 16}$;
5) $\frac{a^3 + b^3}{a^2 - b^2} * \frac{7a - 7b}{a^2 - ab + b^2}$;
6) $\frac{x^2 - 9}{x + y} * \frac{5x + 5y}{x^2 - 3x}$;
7) $\frac{m + 2n}{2 - 3m} : \frac{m^2 + 4mn + 4n^2}{3m^2 - 2m}$;
8) $\frac{a^3 + 8}{16 - a^4} : \frac{a^2 - 2a + 4}{a^2 + 4}$;
9) $\frac{x^2 - 12x + 36}{3x + 21} * \frac{x^2 - 49}{4x - 24}$;
10) $\frac{3a + 15b}{a^2 - 81b^2} : \frac{4a + 20b}{a^2 - 18ab + 81b^2}$.

Решение:

1) $\frac{4 - a}{8a^3} * \frac{12a^5}{a^2 - 16} = \frac{4 - a}{2} * \frac{3a^2}{(a - 4)(a + 4)} = -\frac{a - 4}{2} * \frac{3a^2}{(a - 4)(a + 4)} = -\frac{1}{2} * \frac{3a^2}{a + 4} = -\frac{3a^2}{2(a + 4)}$

2) $\frac{4c - d}{c^2 + cd} * \frac{2c^2 - 2d^2}{4c^2 - cd} = \frac{4c - d}{c(c + d)} * \frac{2(c^2 - d^2)}{c(4c - d)} = \frac{1}{c(c + d)} * \frac{2(c - d)(c + d)}{c} = \frac{1}{c} * \frac{2(c - d)}{c} = \frac{2(c - d)}{c^2}$

3) $\frac{b^2 - 6b + 9}{b^2 - 3b + 9} * \frac{b^3 + 27}{5b - 15} = \frac{(b - 3)^2}{b^2 - 3b + 9} * \frac{(b + 3)(b^2 - 3b + 9)}{5(b - 3)} = \frac{b - 3}{1} * \frac{b + 3}{5} = \frac{b^2 - 9}{5}$

4) $\frac{a^3 - 16a}{3a^2b} * \frac{12ab^2}{4a + 16} = \frac{a(a^2 - 16)}{a} * \frac{4b}{4(a + 4)} = \frac{(a - 4)(a + 4)}{1} * \frac{b}{a + 4} = \frac{a - 4}{1} * \frac{b}{1} = b(a - 4)$

5) $\frac{a^3 + b^3}{a^2 - b^2} * \frac{7a - 7b}{a^2 - ab + b^2} = \frac{(a + b)(a^2 - ab + b^2)}{(a - b)(a + b)} * \frac{7(a - b)}{a^2 - ab + b^2} = \frac{1}{1} * \frac{7}{1} = 7$

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

7) $\frac{m + 2n}{2 - 3m} : \frac{m^2 + 4mn + 4n^2}{3m^2 - 2m} = \frac{m + 2n}{2 - 3m} : \frac{(m + 2n)^2}{m(3m - 2)} = -\frac{m + 2n}{3m - 2n} * \frac{m(3m - 2)}{(m + 2n)^2} = -\frac{1}{1} * \frac{m}{m + 2n} = -\frac{m}{m + 2n}$

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

9) $\frac{x^2 - 12x + 36}{3x + 21} * \frac{x^2 - 49}{4x - 24} = \frac{(x - 6)^2}{3(x + 7)} * \frac{(x - 7)(x + 7)}{4(x - 6)} = \frac{x - 6}{3} * \frac{x - 7}{4} = \frac{(x - 6)(x - 7)}{12}$

10) $\frac{3a + 15b}{a^2 - 81b^2} : \frac{4a + 20b}{a^2 - 18ab + 81b^2} = \frac{3(a + 5b)}{(a - 9b)(a + 9b)} : \frac{4(a + 5b)}{(a - 9b)^2} = \frac{3(a + 5b)}{(a - 9b)(a + 9b)} * \frac{(a - 9b)^2}{4(a + 5b)} = \frac{3}{a + 9b} * \frac{a - 9b}{4} = \frac{3(a - 9b)}{4(a + 9b)}$

159. Упростите выражение:
1) $\frac{7a^2}{a^2 - 25} * \frac{5 - a}{a}$;
2) $\frac{a^3 + b^3}{a^3 - b^3} * \frac{b - a}{b + a}$;
3) $\frac{a^4 - 1}{a^3 - a} * \frac{a}{1 + a^2}$;
4) $\frac{a^2 - 8ab}{12b} : \frac{8b^2 - ab}{24a}$;
5) $\frac{5m^2 - 5n^2}{m^2 + n^2} : \frac{15n - 15m}{4m^2 + 4n^2}$;
6) $\frac{mn^2 - 36m}{m^3 - 8} : \frac{2n + 12}{6m - 12}$;
7) $\frac{a^4 - 1}{a^2 - a + 1} : \frac{a - 1}{a^3 + 1}$;
8) $\frac{4x^2 - 100}{6x} : (2x^2 - 20x + 50)$.

Решение:

1) $\frac{7a^2}{a^2 - 25} * \frac{5 - a}{a} = \frac{7a}{(a - 5)(a + 5)} * \frac{5 - a}{1} = -\frac{7a}{(5 - a)(a + 5)} * \frac{5 - a}{1} = -\frac{7a}{a + 5} * \frac{1}{1} = -\frac{7a}{a + 5}$

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

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

4) $\frac{a^2 - 8ab}{12b} : \frac{8b^2 - ab}{24a} = \frac{a(a - 8b)}{12b} : \frac{b(8b - a)}{24a} = \frac{a(a - 8b)}{12b} * \frac{24a}{b(8b - a)} = \frac{a(a - 8b)}{12b} * (-\frac{24a}{b(a - 8b)}) = \frac{a}{b} * (-\frac{2a}{b}) = -\frac{2a^2}{b^2}$

5) $\frac{5m^2 - 5n^2}{m^2 + n^2} : \frac{15n - 15m}{4m^2 + 4n^2} = \frac{5(m^2 - n^2)}{m^2 + n^2} : \frac{15(n - m)}{4(m^2 + n^2)} = \frac{5(m - n)(m + n)}{m^2 + n^2} : (-\frac{15(m - n)}{4(m^2 + n^2)}) = \frac{5(m - n)(m + n)}{m^2 + n^2} * (-\frac{4(m^2 + n^2)}{15(m - n)}) = \frac{m + n}{1} * (-\frac{4}{3}) = -\frac{4(m + n)}{3}$

6) $\frac{mn^2 - 36m}{m^3 - 8} : \frac{2n + 12}{6m - 12} = \frac{m(n^2 - 36)}{(m - 2)(m^2 + 2m + 4)} : \frac{2(n + 6)}{6(m - 2)} = \frac{m(n - 6)(n + 6)}{(m - 2)(m^2 + 2m + 4)} * \frac{6(m - 2)}{2(n + 6)} = \frac{m(n - 6)}{m^2 + 2m + 4} * \frac{3}{1} = \frac{3m(n - 6)}{m^2 + 2m + 4}$

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

8) $\frac{4x^2 - 100}{6x} : (2x^2 - 20x + 50) = \frac{4(x^2 - 25)}{6x} : 2(x^2 - 10x + 25) = \frac{2(x - 5)(x + 5)}{3x} : 2(x - 5)^2 = \frac{2(x - 5)(x + 5)}{3x} * \frac{1}{2(x - 5)^2} = \frac{x + 5}{3x} * \frac{1}{x - 5} = \frac{x + 5}{3x(x - 5)}$

160. Упростите выражение и найдите его значение:
1) $\frac{a^2 - 81}{a^2 - 8a} : \frac{a - 9}{a^2 - 64}$, если a = −4;
2) $\frac{x}{4x^2 - 4y^2} : \frac{1}{6x + 6y}$, если x = 4,2, y = −2,8;
3) $(3a^2 - 18a + 27) : \frac{3a - 9}{4a}$, если a = 0,5;
4) $\frac{a^6 + a^5}{(3a - 3)^2} : \frac{a^5 + a^4}{9a^2 - 9a}$, если a = 0,8.

Решение:

1) $\frac{a^2 - 81}{a^2 - 8a} : \frac{a - 9}{a^2 - 64} = \frac{(a - 9)(a + 9)}{a(a - 8)} : \frac{a - 9}{(a - 8)(a + 8)} = \frac{(a - 9)(a + 9)}{a(a - 8)} * \frac{(a - 8)(a + 8)}{a - 9} = \frac{a + 9}{a} * \frac{a + 8}{1} = \frac{(a + 9)(a + 8)}{a}$
при a = −4:
$\frac{(a + 9)(a + 8)}{a} = \frac{(-4 + 9)(-4 + 8)}{-4} = \frac{5 * 4}{-4} = -5$

2) $\frac{x}{4x^2 - 4y^2} : \frac{1}{6x + 6y} = \frac{x}{(2x - 2y)(2x + 2y)} : \frac{1}{3(2x + 2y)} = \frac{x}{(2x - 2y)(2x + 2y)} * \frac{3(2x + 2y)}{1} = \frac{x}{2x - 2y} * \frac{3}{1} = \frac{3x}{2(x - y)}$
при x = 4,2, y = −2,8:
$\frac{3x}{2(x - y)} = \frac{3 * 4,2}{2(4,2 - (-2,8))} = \frac{12,6}{2(4,2 + 2,8)} = \frac{12,6}{2 * 7} = \frac{12,6}{14} = \frac{126}{140} = \frac{9}{10} = 0,9$

3) $(3a^2 - 18a + 27) : \frac{3a - 9}{4a} = 3(a^2 - 6a + 9) : \frac{3(a - 3)}{4a} = 3(a - 3)^2 * \frac{4a}{3(a - 3)} = (a - 3) * \frac{4a}{1} = 4a(a - 3)$
при a = 0,5:
4a(a − 3) = 4 * 0,5(0,5 − 3) = 2 * (−2,5) = −5

4) $\frac{a^6 + a^5}{(3a - 3)^2} : \frac{a^5 + a^4}{9a^2 - 9a} = \frac{a^5(a + 1)}{9(a - 1)^2} : \frac{a^4(a + 1)}{9a(a - 1)} = \frac{a^5(a + 1)}{9(a - 1)^2} * \frac{9a(a - 1)}{a^4(a + 1)} = \frac{a}{a - 1} * \frac{a}{1} = \frac{a^2}{a - 1}$
при a = 0,8:
$\frac{a^2}{a - 1} = \frac{0,8^2}{0,8 - 1} = \frac{0,64}{-0,2} = -\frac{64}{20} = -\frac{32}{10} = -3,2$