Ответы к странице 218

860. Выполните умножение:
1) $\frac{2xy - y^2}{9} * \frac{36}{y^4}$;
2) $\frac{a^2 - 7ab}{a^2 + 2ab} * \frac{a^2b + 2ab^2}{a^3 - 7a^2b}$;
3) $\frac{m^2 - 64}{m^3 - 9m^2} * \frac{m^2 - 81}{m^2 + 8m}$;
4) $\frac{2x^2 - 16x + 32}{3x^2 - 6x + 12} * \frac{x^3 + 8}{4x^2 - 64}$.

Решение:

1) $\frac{2xy - y^2}{9} * \frac{36}{y^4} = \frac{y(2x - y)}{1} * \frac{4}{y^4} = \frac{4(2x - y)}{y^3}$

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

3) $\frac{m^2 - 64}{m^3 - 9m^2} * \frac{m^2 - 81}{m^2 + 8m} = \frac{(m - 8)(m + 8)}{m^2(m - 9)} * \frac{(m - 9)(m + 9)}{m(m + 8)} = \frac{m - 8}{m^2} * \frac{m + 9}{m} = \frac{(m - 8)(m + 9)}{m^3}$

4) $\frac{2x^2 - 16x + 32}{3x^2 - 6x + 12} * \frac{x^3 + 8}{4x^2 - 64} = \frac{2(x^2 - 8x + 16)}{3(x^2 - 2x + 4)} * \frac{(x + 2)(x^2 - 2x + 4)}{4(x^2 - 16)} = \frac{2(x - 4)^2}{3} * \frac{x + 2}{4(x - 4)(x + 4)} = \frac{x - 4}{3} * \frac{x + 2}{2(x + 4)} = \frac{(x - 4)(x + 2)}{6(x + 4)}$

861. Представьте выражение в виде дроби:
1) $(\frac{a^5}{x^4})^2$;
2) $(-\frac{4y}{3m^2})^4$;
3) $(-\frac{10x^2y^5}{3a^4b^3})^3$;
4) $(-\frac{2a^4b^4}{25x^5})^2 * (-\frac{5x^2}{4a^2b^3})^3$.

Решение:

1) $(\frac{a^5}{x^4})^2 = \frac{(a^5)^2}{(x^4)^2} = \frac{a^{10}}{x^{8}}$

2) $(-\frac{4y}{3m^2})^4 = \frac{(4y)^4}{(3m^2)^4} = \frac{256y^4}{81m^8}$

3) $(-\frac{10x^2y^5}{3a^4b^3})^3 = -\frac{(10x^2y^5)^3}{(3a^4b^3)^3} = -\frac{1000x^6y^{15}}{27a^{12}b^9}$

4) $(-\frac{2a^4b^4}{25x^5})^2 * (-\frac{5x^2}{4a^2b^3})^3 = \frac{(2a^4b^4)^2}{(25x^5)^2} * (-\frac{(5x^2)^3}{(4a^2b^3)^3}) = \frac{4a^8b^8}{625x^{10}} * (-\frac{125x^6}{64a^6b^9}) = \frac{a^2}{5x^{4}} * (-\frac{1}{16b}) = -\frac{a^2}{80bx^{4}}$

862. Выполните деление:
1) $\frac{x^2 - 10x + 25}{x^2 - 100} : \frac{x - 5}{x - 10}$;
2) $\frac{a^2 - 1}{a - 8} : \frac{a^2 + 2a + 1}{a - 8}$;
3) $\frac{ab + b^2}{8b} : \frac{ab + a^2}{2a}$;
4) $\frac{2c - 3}{c - 1} : (2c - 3)$;
5) $\frac{x^2 - 16y^2}{25x^2 - 4y^2} : \frac{x^2 + 8xy + 16y^2}{25x^2 + 20xy + 4y^2}$;
6) $\frac{n^2 - 3n}{49n^2 - 1} : \frac{n^4 - 27n}{49n^2 - 14n + 1}$;
7) $\frac{m^{12} - n^{15}}{2m^{10} - 8n^{14}} : \frac{5m^8 + 5m^4n^5 + 5n^{10}}{3m^5 + 6n^7}$;
8) $\frac{5a^2 - 20ab}{3a^2 + b^2} : \frac{30(a - 4b)^2}{9a^4 - b^4}$.

Решение:

1) $\frac{x^2 - 10x + 25}{x^2 - 100} : \frac{x - 5}{x - 10} = \frac{(x - 5)^2}{(x - 10)(x + 10)} * \frac{x - 10}{x - 5} = \frac{x - 5}{x + 10}$

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

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

4) $\frac{2c - 3}{c - 1} : (2c - 3) = \frac{2c - 3}{c - 1} * \frac{1}{2c - 3} = \frac{1}{c - 1}$

5) $\frac{x^2 - 16y^2}{25x^2 - 4y^2} : \frac{x^2 + 8xy + 16y^2}{25x^2 + 20xy + 4y^2} = \frac{(x - 4y)(x + 4y)}{(5x - 2y)(5x + 2y)} : \frac{(x + 4y)^2}{(5x + 2y)^2} = \frac{(x - 4y)(x + 4y)}{(5x - 2y)(5x + 2y)} * \frac{(5x + 2y)^2}{(x + 4y)^2} = \frac{x - 4y}{5x - 2y} * \frac{5x + 2y}{x + 4y} = \frac{(x - 4y)(5x + 2y)}{(x + 4y)(5x - 2y)}$

6) $\frac{n^2 - 3n}{49n^2 - 1} : \frac{n^4 - 27n}{49n^2 - 14n + 1} = \frac{n(n - 3)}{(7n - 1)(7n + 1)} : \frac{n(n^3 - 27)}{(7n - 1)^2} = \frac{n(n - 3)}{(7n - 1)(7n + 1)} * \frac{(7n - 1)^2}{n(n^3 - 27)} = \frac{n - 3}{7n + 1} * \frac{7n - 1}{n^3 - 27} = \frac{n - 3}{7n + 1} * \frac{7n - 1}{(n - 3)(n^2 + 3n + 9)} = \frac{1}{7n + 1} * \frac{7n - 1}{n^2 + 3n + 9} = \frac{7n - 1}{(7n + 1)(n^2 + 3n + 9)}$

7) $\frac{m^{12} - n^{15}}{2m^{10} - 8n^{14}} : \frac{5m^8 + 5m^4n^5 + 5n^{10}}{3m^5 + 6n^7} = \frac{(m^{4})^3 - (n^{5})^3}{2(m^{10} - 4n^{14})} : \frac{5(m^8 + m^4n^5 + n^{10})}{3(m^5 + 2n^7)} = \frac{(m^{4} - n^{5})(m^8 + m^4n^5 + n^{10})}{2(m^{5} - 2n^{7})(m^5 + 2n^7)} * \frac{3(m^5 + 2n^7)}{5(m^8 + m^4n^5 + n^{10})} = \frac{m^{4} - n^{5}}{2(m^{5} - 2n^{7})} * \frac{3}{5} = \frac{3(m^{4} - n^{5})}{10(m^{5} - 2n^{7})}$

8) $\frac{5a^2 - 20ab}{3a^2 + b^2} : \frac{30(a - 4b)^2}{9a^4 - b^4} = \frac{5a(a - 4b)}{3a^2 + b^2} : \frac{30(a - 4b)^2}{(3a^2 - b^2)(3a^2 + b^2)} = \frac{5a(a - 4b)}{3a^2 + b^2} * \frac{(3a^2 - b^2)(3a^2 + b^2)}{30(a - 4b)^2} = \frac{a}{1} * \frac{3a^2 - b^2}{6(a - 4b)} = \frac{a(3a^2 - b^2)}{6(a - 4b)}$

863. Полагая данные дроби несократимыми, замените x и y такими одночленами, чтобы получилось тождество:
1) $\frac{x}{7a^2b^3} * \frac{y}{4c} = \frac{6a^3c^2}{b}$;
2) $\frac{36m^2n^4}{x} : \frac{y}{35p^6} = \frac{21n}{5mp^3}$.

Решение:

1) $\frac{x}{7a^2b^3} * \frac{y}{4c} = \frac{6a^3c^2}{b}$
$\frac{24c^3}{7a^2b^3} * \frac{7a^5b^2}{4c} = \frac{6a^3c^2}{b}$
Ответ: $x = 24c^3, y = 7a^5b^2$.

2) $\frac{36m^2n^4}{x} : \frac{y}{35p^6} = \frac{21n}{5mp^3}$
$\frac{36m^2n^4}{x} * \frac{35p^6}{y} = \frac{21n}{5mp^3}$
$\frac{36m^2n^4}{25p^9} * \frac{35p^6}{12m^3n^3} = \frac{21n}{5mp^3}$
Ответ: $x = 25p^9, y = 12m^3n^3$.

864. Дано: $3x - \frac{1}{x} = 8$. Найдите значение выражения $9x^2 + \frac{1}{x^2}$.

Решение:

$3x - \frac{1}{x} = 8$
$(3x - \frac{1}{x})^2 = 8^2$
$9x^2 - 2 * 3x * \frac{1}{x} + (\frac{1}{x})^2 = 64$
$9x^2 - 6 + \frac{1}{x^2} = 64$
$9x^2 + \frac{1}{x^2} = 64 + 6$
$9x^2 + \frac{1}{x^2} = 70$
Ответ: 70

865. Дано: $4x^2 + \frac{1}{x^2} = 6$. Найдите значение выражения $2x - \frac{1}{x}$.

Решение:

$4x^2 + \frac{1}{x^2} = 6$
$4x^2 + \frac{1}{x^2} - 4x * \frac{1}{x} + 4 = 6$
$4x^2 - 4x * \frac{1}{x} + \frac{1}{x^2} + 4 = 6$
$(2x - \frac{1}{x})^2 = 6 - 4$
$(2x - \frac{1}{x})^2 = 2$
$2x - \frac{1}{x} = ±\sqrt{2}$
Ответ: $-\sqrt{2}$ или $\sqrt{2}$

866. Упростите выражение:
1) $\frac{x^{3k}}{y^{2n}} : \frac{x^{6k}}{y^{5n}}$, где k и n − целые числа;
2) $\frac{a^{k + 5} * b^{k + 3}}{c^{3k + 2}} : \frac{a^{k + 3} * b^{k + 2}}{c^{2k + 1}}$, где k − целое число;
3) $\frac{(x^n + 3y^n)^2 - 12x^ny^n}{x^{3n} + 27y^{3n}} : \frac{x^{2n} - 9y^{2n}}{(x^n - 3y^n)^2 + 12x^ny^n}$, где n − целое число.

Решение:

1) $\frac{x^{3k}}{y^{2n}} : \frac{x^{6k}}{y^{5n}} = \frac{x^{3k}}{y^{2n}} * \frac{y^{5n}}{x^{6k}} = \frac{y^{5n - 2n}}{x^{6k - 3k}} = \frac{y^{3n}}{x^{3k}}$

2) $\frac{a^{k + 5} * b^{k + 3}}{c^{3k + 2}} : \frac{a^{k + 3} * b^{k + 2}}{c^{2k + 1}} = \frac{a^{k + 5} * b^{k + 3}}{c^{3k + 2}} * \frac{c^{2k + 1}}{a^{k + 3} * b^{k + 2}} = \frac{a^{k + 5 - (k + 3)} * b^{k + 3 - (k + 2)}}{c^{3k + 2 - (2k + 1)}} = \frac{a^{k + 5 - k - 3} * b^{k + 3 - k - 2}}{c^{3k + 2 - 2k - 1}} = \frac{a^{2}b}{c^{k + 1}}$

3) $\frac{(x^n + 3y^n)^2 - 12x^ny^n}{x^{3n} + 27y^{3n}} : \frac{x^{2n} - 9y^{2n}}{(x^n - 3y^n)^2 + 12x^ny^n} = \frac{(x^n + 3y^n)^2 - 12x^ny^n}{x^{3n} + 27y^{3n}} * \frac{(x^n - 3y^n)^2 + 12x^ny^n}{x^{2n} - 9y^{2n}} = \frac{x^{2n} + 6x^ny^n + 9y^{2n} - 12x^ny^n}{(x^n + 3y^n)(x^{2n - 3x^ny^n + 9y^{2n}})} * \frac{x^{2n} - 6x^ny^n + 9y^{2n} + 12x^ny^n}{(x^n - 3y^n)(x^n + 3y^n)} = \frac{x^{2n} - 6x^ny^n + 9y^{2n}}{(x^n + 3y^n)(x^{2n} - 3x^ny^n + 9y^{2n})} * \frac{x^{2n} + 6x^ny^n + 9y^{2n}}{(x^n - 3y^n)(x^n + 3y^n)} = \frac{(x^n - 3y^n)^2}{(x^n + 3y^n)(x^{2n} - 3x^ny^n + 9y^{2n})} * \frac{(x^n + 3y^n)^2}{(x^n - 3y^n)(x^n + 3y^n)} = \frac{x^n - 3y^n}{x^{2n} - 3x^ny^n + 9y^{2n}}$

867. Упростите выражение:
1) $(\frac{a + 4}{a - 4} - \frac{a - 4}{a + 4}) * \frac{16 - a^2}{32a^3}$;
2) $(7x - \frac{4x}{x - 3}) : \frac{14x - 50}{3x - 9}$;
3) $\frac{2a}{a - 2} + \frac{a + 7}{8 - 4a} * \frac{32}{7a + a^2}$;
4) $(\frac{9c}{c - 8} + \frac{7c}{c^2 - 16c + 64}) : \frac{9c - 65}{c^2 - 64} - \frac{8c + 64}{c - 8}$;
5) $(\frac{a^2}{a + b} - \frac{a^3}{a^2 + ab + b^2}) : (\frac{a}{a - b} - \frac{a^2}{a^2 - b^2})$;
6) $(\frac{b}{b + 6} + \frac{36 + b^2}{36 - b^2} - \frac{b}{b - 6}) : \frac{6b + b^2}{(6 - b)^2}$;
7) $(\frac{2x}{x^3 + 1} : \frac{1 - x}{x^2 - x + 1} + \frac{2}{x - 1}) * \frac{x^2 - 2x + 1}{4} : \frac{x - 1}{x + 1}$.

Решение:

1) $(\frac{a + 4}{a - 4} - \frac{a - 4}{a + 4}) * \frac{16 - a^2}{32a^3} = \frac{(a + 4)^2 - (a - 4)^2}{(a - 4)(a + 4)} * \frac{16 - a^2}{32a^3} = \frac{a^2 + 8a + 16 - (a^2 - 8a + 16)}{a^2 - 16} * \frac{16 - a^2}{32a^3} = \frac{a^2 + 8a + 16 - a^2 + 8a - 16}{a^2 - 16} * (-\frac{a^2 - 16}{32a^3}) = \frac{16a}{1} * (-\frac{1}{32a^3}) = -\frac{1}{2a^2}$

2) $(7x - \frac{4x}{x - 3}) : \frac{14x - 50}{3x - 9} = \frac{7x(x - 3) - 4x}{x - 3} : \frac{2(7x - 25)}{3(x - 3)} = \frac{7x(x - 3) - 4x}{x - 3} * \frac{3(x - 3)}{2(7x - 25)} = \frac{7x^2 - 21x - 4x}{1} * \frac{3}{2(7x - 25)} = \frac{7x^2 - 25x}{1} * \frac{3}{2(7x - 25)} = \frac{x(7x - 25)}{1} * \frac{3}{2(7x - 25)} = \frac{3x}{2}$

3) $\frac{2a}{a - 2} + \frac{a + 7}{8 - 4a} * \frac{32}{7a + a^2} = \frac{2a}{a - 2} + \frac{a + 7}{4(2 - a)} * \frac{32}{a(7 + a)} = \frac{2a}{a - 2} + \frac{1}{2 - a} * \frac{8}{a} = \frac{2a}{a - 2} - \frac{8}{a(a - 2)} = \frac{2a^2 - 8}{a(a - 2)} = \frac{2(a^2 - 4)}{a(a - 2)} = \frac{2(a - 2)(a + 2)}{a(a - 2)} = \frac{2(a + 2)}{a}$

4) $(\frac{9c}{c - 8} + \frac{7c}{c^2 - 16c + 64}) : \frac{9c - 65}{c^2 - 64} - \frac{8c + 64}{c - 8} = (\frac{9c}{c - 8} + \frac{7c}{(c - 8)^2}) : \frac{9c - 65}{(c - 8)(c + 8)} - \frac{8(c + 8)}{c - 8} = \frac{9c(c - 8) + 7c}{(c - 8)^2} * \frac{(c - 8)(c + 8)}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{9c^2 - 72c + 7c}{c - 8} * \frac{c + 8}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{9c^2 - 65c}{c - 8} * \frac{c + 8}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{c(9c - 65)}{c - 8} * \frac{c + 8}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{c}{c - 8} * \frac{c + 8}{1} - \frac{8(c + 8)}{c - 8} = \frac{c(c + 8)}{c - 8} - \frac{8(c + 8)}{c - 8} = \frac{c(c + 8) - 8(c + 8)}{c - 8} = \frac{(c + 8)(c - 8)}{c - 8} = c + 8$

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

6) $(\frac{b}{b + 6} + \frac{36 + b^2}{36 - b^2} - \frac{b}{b - 6}) : \frac{6b + b^2}{(6 - b)^2} = (\frac{b}{b + 6} - \frac{36 + b^2}{b^2 - 36} - \frac{b}{b - 6}) : \frac{b(6 + b)}{(6 - b)^2} = (\frac{b}{b + 6} - \frac{36 + b^2}{(b - 6)(b + 6)} - \frac{b}{b - 6}) : \frac{b(6 + b)}{(6 - b)^2} = \frac{b(b - 6) - (36 + b^2) - b(b + 6)}{(b - 6)(b + 6)} * \frac{(6 - b)^2}{b(6 + b)} = \frac{b^2 - 6b - 36 - b^2 - b^2 - 6b}{b + 6} * \frac{6 - b}{b(6 + b)} = \frac{-b^2 - 12b - 36}{b + 6} * \frac{6 - b}{b(6 + b)} = \frac{-(b^2 + 12b + 36)}{b + 6} * \frac{6 - b}{b(6 + b)} = \frac{-(b + 6)^2}{b + 6} * \frac{6 - b}{b(6 + b)} = -\frac{6 - b}{b}$

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