Ответы к странице 216

845. Сократите дробь (n − натуральное число):
1) $\frac{100^n}{2^{2n + 3} * 5^{2n + 1}}$;
2) $\frac{2^{2n + 1} * 7^{n + 1}}{6 * 28^n}$;
3) $\frac{5^{n + 1} - 5^n}{2 * 5^n}$;
4) $\frac{18^n}{3^{2n + 2} * 2^{n + 3}}$;
5) $\frac{41 * 9^n}{9^{n + 2} + 9^n}$.

Решение:

1) $\frac{100^n}{2^{2n + 3} * 5^{2n + 1}} = \frac{(4 * 25)^n}{2^{2n + 3} * 5^{2n + 1}} = \frac{4^n * 25^n}{2^{2n + 3} * 5^{2n + 1}} = \frac{(2^2)^n * (5^2)^n}{2^{2n + 3} * 5^{2n + 1}} = \frac{2^{2n} * 5^{2n}}{2^{2n + 3} * 5^{2n + 1}} = 2^{2n - (2n + 3)} * 5^{2n - (2n + 1)} = 2^{2n - 2n - 3} * 5^{2n - 2n - 1} = 2^{-3} * 5^{-1} = \frac{1}{2^3 * 5} = \frac{1}{8 * 5} = \frac{1}{40}$

2) $\frac{2^{2n + 1} * 7^{n + 1}}{6 * 28^n} = \frac{2^{2n + 1} * 7^{n + 1}}{2 * 3 * (4 * 7)^n} = \frac{2^{2n + 1} * 7^{n + 1}}{2 * 3 * 4^n * 7^n} = \frac{2^{2n + 1} * 7^{n + 1}}{2 * 3 * (2^2)^n * 7^n} = \frac{2^{2n + 1} * 7^{n + 1}}{2 * 3 * 2^{2n} * 7^n} = \frac{2^{2n + 1} * 7^{n + 1}}{3 * 2^{2n + 1} * 7^n} = \frac{7^{n + 1}}{3 * 7^n} = \frac{7^{n + 1 - n}}{3} = \frac{7}{3} = 2\frac{1}{3}$

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

4) $\frac{18^n}{3^{2n + 2} * 2^{n + 3}} = \frac{(9 * 2)^n}{3^{2n + 2} * 2^{n + 3}} = \frac{9^n * 2^n}{3^{2n + 2} * 2^{n + 3}} = \frac{(3^2)^n * 2^n}{3^{2n + 2} * 2^{n + 3}} = \frac{3^{2n} * 2^n}{3^{2n + 2} * 2^{n + 3}} = 3^{2n - (2n + 2)} * 2^{n - (n + 3)} = 3^{2n - 2n - 2} * 2^{n - n - 3} = 3^{-2} * 2^{-3} = \frac{1}{3^2 * 2^3} = \frac{1}{9 * 8} = \frac{1}{72}$

5) $\frac{41 * 9^n}{9^{n + 2} + 9^n} = \frac{41 * 9^n}{9^{n}(9^2 + 1)} = \frac{41}{81 + 1} = \frac{41}{82} = \frac{1}{2}$

846. Для каждого значения a решите уравнение:
1) (a + 2)x = 7;
2) (a + 6)x = a + 6;
3) $(a + 3)x = a^2 + 6a + 9$;
4) $(a^2 - 4)x = a - 2$.

Решение:

1) (a + 2)x = 7
если a ≠ −2:
$x = \frac{7}{a + 2}$
если a = −2:
(−2 + 2)x = 7
0x = 7
0 ≠ 7
нет корней
Ответ:
если a ≠ −2: $x = \frac{7}{a + 2}$;
если a = −2: нет корней.

2) (a + 6)x = a + 6
если a ≠ −6:
$x = \frac{a + 6}{a + 6}$
x = 1
если a = −6:
(−6 + 6)x = −6 + 6
0x = 0
0 = 0
x − любое число
Ответ:
если a ≠ −6: x = 1;
если a = −6: x − любое число.

3) $(a + 3)x = a^2 + 6a + 9$
$(a + 3)x = (a + 3)^2$
если a ≠ −3:
$x = \frac{(a + 3)^2}{a + 3}$
x = a + 3
если a = −3:
$(-3 + 3)x = (-3 + 3)^2$
0x = 0
0 = 0
x − любое число
Ответ:
если a ≠ −3: x = a + 3;
если a = −3: x − любое число.

4) $(a^2 - 4)x = a - 2$
$(a - 2)(a + 2)x = a - 2$
если a ≠ −2 и a ≠ 2:
$x = \frac{a - 2}{(a - 2)(a + 2)}$
$x = \frac{1}{a + 2}$
если a = −2:
$(-2 - 2)(-2 + 2)x = -2 - 2$
−4 * 0x = −4
0 ≠ −4
нет корней
если a = 2:
$(2 - 2)(2 + 2)x = 2 - 2$
0 * 4x = 0
0 = 0
x − любое число
Ответ:
если a ≠ −2 и a ≠ 2: $x = \frac{1}{a + 2}$;
если a = −2: нет корней;
если a = 2: x − любое число.

847. Представьте в виде дроби выражение:
1) $\frac{7a}{22} + \frac{4a}{22}$;
2) $\frac{8x}{3y} - \frac{5x}{3y}$;
3) $\frac{7x - 2y}{15p} + \frac{3x + 7y}{15p}$;
4) $\frac{x + y}{9p} - \frac{x}{9p}$;
5) $\frac{a}{8} - \frac{a - b}{8}$;
6) $\frac{7p - 17}{5k} + \frac{7 - 2p}{5k}$;
7) $\frac{6a^2 - 4a}{15a} - \frac{a^2 + a}{15a}$;
8) $\frac{x - y}{8} + \frac{x + y}{8}$;
9) $\frac{10x - 6}{x} - \frac{4x + 11}{x}$.

Решение:

1) $\frac{7a}{22} + \frac{4a}{22} = \frac{7a + 4a}{22} = \frac{11a}{22} = \frac{a}{2}$

2) $\frac{8x}{3y} - \frac{5x}{3y} = \frac{8x - 5x}{3y} = \frac{3x}{3y} = \frac{x}{y}$

3) $\frac{7x - 2y}{15p} + \frac{3x + 7y}{15p} = \frac{7x - 2y + 3x + 7y}{15p} = \frac{10x + 5y}{15p} = \frac{5(2x + y)}{15p} = \frac{2x + y}{3p}$

4) $\frac{x + y}{9p} - \frac{x}{9p} = \frac{x + y - x}{9p} = \frac{y}{9p}$

5) $\frac{a}{8} - \frac{a - b}{8} = \frac{a - (a - b)}{8} = \frac{a - a + b}{8} = \frac{b}{8}$

6) $\frac{7p - 17}{5k} + \frac{7 - 2p}{5k} = \frac{7p - 17 + 7 - 2p}{5k} = \frac{5p - 10}{5k} = \frac{5(p - 2)}{5k} = \frac{p - 2}{k}$

7) $\frac{6a^2 - 4a}{15a} - \frac{a^2 + a}{15a} = \frac{6a^2 - 4a - (a^2 + a)}{15a} = \frac{6a^2 - 4a - a^2 - a}{15a} = \frac{5a^2 - 5a}{15a} = \frac{5a(a - 1)}{15a} = \frac{a - 1}{3}$

8) $\frac{x - y}{8} + \frac{x + y}{8} = \frac{x - y + x + y}{8} = \frac{2x}{8} = \frac{x}{4}$

9) $\frac{10x - 6}{x} - \frac{4x + 11}{x} = \frac{10x - 6 - (4x + 11)}{x} = \frac{10x - 6 - 4x - 11}{x} = \frac{6x - 17}{x}$

848. Упростите выражение:
1) $\frac{7y}{y^2 - 4} - \frac{14}{y^2 - 4}$;
2) $\frac{y^2 - 3y}{25 - y^2} - \frac{7y - 25}{25 - y^2}$;
3) $\frac{9p + 5}{3p + 6} - \frac{10p - 12}{3p + 6} + \frac{9p - 1}{3p + 6}$;
4) $\frac{7x + 5}{3 - x} + \frac{5x + 11}{x - 3}$;
5) $\frac{(3a - 1)^2}{4a - 4} + \frac{(a - 3)^2}{4 - 4a}$;
6) $\frac{x^2 - 3x}{(2 - x)^2} - \frac{x - 4}{(x - 2)^2}$;
7) $\frac{7}{a - 2} - \frac{b}{2 - a}$;
8) $\frac{6a}{5 - a} - \frac{4a}{a - 5}$.

Решение:

1) $\frac{7y}{y^2 - 4} - \frac{14}{y^2 - 4} = \frac{7y - 14}{y^2 - 4} = \frac{7(y - 2)}{(y - 2)(y + 2)} = \frac{7}{y + 2}$

2) $\frac{y^2 - 3y}{25 - y^2} - \frac{7y - 25}{25 - y^2} = \frac{y^2 - 3y - (7y - 25)}{25 - y^2} = \frac{y^2 - 3y - 7y + 25}{25 - y^2} = \frac{y^2 - 10y + 25}{25 - y^2} = \frac{(y - 5)^2}{(5 - y)(5 + y)} = \frac{(5 - y)^2}{(5 - y)(5 + y)} = \frac{5 - y}{5 + y}$

3) $\frac{9p + 5}{3p + 6} - \frac{10p - 12}{3p + 6} + \frac{9p - 1}{3p + 6} = \frac{9p + 5 - (10p - 12) + 9p - 1}{3p + 6} = \frac{9p + 5 - 10p + 12 + 9p - 1}{3p + 6} = \frac{8p + 16}{3p + 6} = \frac{8(p + 2)}{3(p + 2)} = \frac{8}{3} = 2\frac{2}{3}$

4) $\frac{7x + 5}{3 - x} + \frac{5x + 11}{x - 3} = \frac{7x + 5}{3 - x} - \frac{5x + 11}{3 - x} = \frac{7x + 5 - (5x + 11)}{3 - x} = \frac{7x + 5 - 5x - 11}{3 - x} = \frac{2x - 6}{3 - x} = \frac{2(x - 3)}{3 - x} = -\frac{2(x - 3)}{x - 3} = -2$

5) $\frac{(3a - 1)^2}{4a - 4} + \frac{(a - 3)^2}{4 - 4a} = \frac{(3a - 1)^2}{4a - 4} - \frac{(a - 3)^2}{4a - 4} = \frac{(3a - 1)^2 - (a - 3)^2}{4a - 4} = \frac{9a^2 - 6a + 1 - (a^2 - 6a + 9)}{4a - 4} = \frac{9a^2 - 6a + 1 - a^2 + 6a - 9}{4a - 4} = \frac{8a^2 - 8}{4a - 4} = \frac{8(a^2 - 1)}{4(a - 1)} = \frac{8(a - 1)(a + 1)}{4(a - 1)} = 2(a + 1)$

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

7) $\frac{7}{a - 2} - \frac{b}{2 - a} = \frac{7}{a - 2} + \frac{b}{a - 2} = \frac{7 + b}{a - 2}$

8) $\frac{6a}{5 - a} - \frac{4a}{a - 5} = \frac{6a}{5 - a} + \frac{4a}{5 - a} = \frac{6a + 4a}{5 - a} = \frac{10a}{5 - a}$

849. Выполните действия:
1) $\frac{8}{x} - \frac{5}{y}$;
2) $\frac{7}{ab} + \frac{5}{b}$;
3) $\frac{5}{24xy} - \frac{7}{18xy}$;
4) $\frac{5b^2 - 8b + 1}{a^2b^2} - \frac{2b - 1}{a^2b}$.

Решение:

1) $\frac{8}{x} - \frac{5}{y} = \frac{8y - 5x}{xy}$

2) $\frac{7}{ab} + \frac{5}{b} = \frac{7 + 5a}{ab}$

3) $\frac{5}{24xy} - \frac{7}{18xy} = \frac{5 * 3 - 7 * 4}{72xy} = \frac{15 - 28}{72xy} = \frac{-13}{72xy} = -\frac{13}{72xy}$

4) $\frac{5b^2 - 8b + 1}{a^2b^2} - \frac{2b - 1}{a^2b} = \frac{5b^2 - 8b + 1 - b(2b - 1)}{a^2b^2} = \frac{5b^2 - 8b + 1 - 2b^2 + b}{a^2b^2} = \frac{3b^2 - 7b + 1}{a^2b^2}$

850. Выполните действия:
1) $\frac{2a - 1}{a - 4} - \frac{3a + 2}{2(a - 4)}$;
2) $\frac{x + 2}{3x + 9} - \frac{4 - x}{5x + 15}$;
3) $\frac{m + 1}{m - 3} - \frac{m + 2}{m + 3}$;
4) $\frac{x}{x + y} - \frac{2y^2}{y^2 - x^2} - \frac{y}{x - y}$;
5) $\frac{m}{3m - 2n} - \frac{3m^2 - 3mn}{9m^2 - 12m + 4n^2}$;
6) $\frac{a + 3}{a^2 - 2a} - \frac{a - 2}{5a - 10} + \frac{a + 2}{5a}$;
7) $\frac{3}{3a - 3} - \frac{a - 1}{2a^2 - 4a + 2}$;
8) $2 - \frac{14}{m - 2} - m$;
9) $\frac{2x + 1}{x^2 - 6x + 9} - \frac{8}{x^2 - 9} - \frac{2x - 1}{x^2 + 6x + 9}$.

Решение:

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

2) $\frac{x + 2}{3x + 9} - \frac{4 - x}{5x + 15} = \frac{x + 2}{3(x + 3)} - \frac{4 - x}{5(x + 3)} = \frac{5(x + 2) - 3(4 - x)}{15(x + 3)} = \frac{5x + 10 - 12 + 3x}{15(x + 3)} = \frac{8x - 2}{15(x + 3)}$

3) $\frac{m + 1}{m - 3} - \frac{m + 2}{m + 3} = \frac{(m + 1)(m + 3) - (m + 2)(m - 3)}{(m - 3)(m + 3)} = \frac{m^2 + m + 3m + 3 - (m^2 + 2m - 3m - 6)}{m^2 - 9} = \frac{m^2 + 4m + 3 - m^2 - 2m + 3m + 6}{m^2 - 9} = \frac{5m + 9}{m^2 - 9}$

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

5) $\frac{m}{3m - 2n} - \frac{3m^2 - 3mn}{9m^2 - 12m + 4n^2} = \frac{m}{3m - 2n} - \frac{3m^2 - 3mn}{(3m - 2n)^2} = \frac{m(3m - 2n) - (3m^2 - 3mn)}{(3m - 2n)^2} = \frac{3m^2 - 2mn - 3m^2 + 3mn}{(3m - 2n)^2} = \frac{mn}{(3m - 2n)^2}$

6) $\frac{a + 3}{a^2 - 2a} - \frac{a - 2}{5a - 10} + \frac{a + 2}{5a} = \frac{a + 3}{a(a - 2)} - \frac{a - 2}{5(a - 2)} + \frac{a + 2}{5a} = \frac{5(a + 3) - a(a - 2) + (a - 2)(a + 2)}{5a(a - 2)} = \frac{5a + 15 - a^2 + 2a + a^2 - 4}{5a(a - 2)} = \frac{7a + 11}{5a(a - 2)}$

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

8) $2 - \frac{14}{m - 2} - m = \frac{2(m - 2) - 14 - m(m - 2)}{m - 2} = \frac{2m - 4 - 14 - m^2 + 2m}{m - 2} = \frac{-m^2 + 4m - 18}{m - 2}$

9) $\frac{2x + 1}{x^2 - 6x + 9} - \frac{8}{x^2 - 9} - \frac{2x - 1}{x^2 + 6x + 9} = \frac{2x + 1}{(x - 3)^2} - \frac{8}{(x - 3)(x + 3)} - \frac{2x - 1}{(x + 3)^2} = \frac{(2x + 1)(x + 3)^2 - 8(x - 3)(x + 3) - (2x - 1)(x - 3)^2}{(x - 3)^2(x + 3)^2} = \frac{(2x + 1)(x^2 + 6x + 9) - 8(x^2 - 9) - (2x - 1)(x^2 - 6x + 9)}{(x - 3)^2(x + 3)^2} = \frac{2x^3 + 12x^2 + 18x + x^2 + 6x + 9 - 8x^2 + 72 - (2x^3 - 12x^2 + 18x - x^2 + 6x - 9)}{(x - 3)^2(x + 3)^2} = \frac{18x^2 + 90}{(x - 3)^2(x + 3)^2}$