Ответы к странице 37

144. Выполните умножение:
1) $\frac{3a^2}{c} * \frac{a^2}{c}$;
2) $\frac{2a}{b} * \frac{b}{8a}$;
3) $\frac{x}{yz} * \frac{y^4}{5x}$;
4) $\frac{3m}{16n^2} * 8n^6$;
5) $14m^9 * \frac{n^2}{7m^3}$;
6) $\frac{15a^4}{b^{12}} * \frac{b^6}{10a^2}$;
7) $\frac{48ab}{17c^4} * \frac{51bc^5}{40a^4}$;
8) $\frac{21c^3}{13p^2} * \frac{39p}{28c^2}$.

Решение:

1) $\frac{3a^2}{c} * \frac{a^2}{c} = \frac{3a^2 * a^2}{c * c} = \frac{3a^4}{c^2}$

2) $\frac{2a}{b} * \frac{b}{8a} = \frac{1}{1} * \frac{1}{4} = \frac{1}{4}$

3) $\frac{x}{yz} * \frac{y^4}{5x} = \frac{1}{z} * \frac{y^3}{5} = \frac{y^3}{5z}$

4) $\frac{3m}{16n^2} * 8n^6 = \frac{3m}{2} * n^4 = \frac{3mn^4}{2}$

5) $14m^9 * \frac{n^2}{7m^3} = 2m^6 * \frac{n^2}{1} = 2m^6n^2$

6) $\frac{15a^4}{b^{12}} * \frac{b^6}{10a^2} = \frac{3a^2}{b^{6}} * \frac{1}{2} = \frac{3a^2}{2b^{6}}$

7) $\frac{48ab}{17c^4} * \frac{51bc^5}{40a^4} = \frac{6b}{1} * \frac{3bc}{5a^3} = \frac{18b^2c}{5a^3}$

8) $\frac{21c^3}{13p^2} * \frac{39p}{28c^2} = \frac{3c}{p} * \frac{3}{4} = \frac{9c}{4p}$

145. Упростите выражение:
1) $\frac{a^2}{b^6} * \frac{b^2}{a^2}$;
2) $\frac{4m^2}{k^5} * \frac{mk^5}{12}$;
3) $\frac{a}{2b} * 2a$;
4) $15x^{12} * \frac{y^2}{5x^4}$;
5) $\frac{11x^3}{y^8} * \frac{y^5}{33x^7}$;
6) $\frac{7k^8}{9mp} * \frac{27m^3}{56k^6p^2}$.

Решение:

1) $\frac{a^2}{b^6} * \frac{b^2}{a^2} = \frac{1}{b^4} * \frac{1}{1} = \frac{1}{b^4}$

2) $\frac{4m^2}{k^5} * \frac{mk^5}{12} = \frac{m^2}{1} * \frac{m}{3} = \frac{m^3}{3}$

3) $\frac{a}{2b} * 2a = \frac{a}{b} * a = \frac{a^2}{b}$

4) $15x^{12} * \frac{y^2}{5x^4} = 3x^{8} * \frac{y^2}{1} = 3x^{8}y^2$

5) $\frac{11x^3}{y^8} * \frac{y^5}{33x^7} = \frac{1}{y^3} * \frac{1}{3x^4} = \frac{1}{3x^4y^3}$

6) $\frac{7k^8}{9mp} * \frac{27m^3}{56k^6p^2} = \frac{k^2}{p} * \frac{3m^2}{8p^2} = \frac{3k^2m^2}{8p^3}$

146. Упростите выражение:
1) $\frac{a - b}{3b} * \frac{3}{a - b}$;
2) $\frac{2mn + n^2}{6m} * \frac{2m}{n}$;
3) $\frac{7a + 7b}{b^6} * \frac{b^3}{a + b}$;
4) $\frac{32a}{a^2 - 9} * \frac{a - 3}{8a}$;
5) $\frac{c - 1}{c + 6} * \frac{c + 6}{c^2 - 2c + 1}$;
6) $\frac{m - 2}{m^2 - 49} * \frac{m + 7}{m - 2}$;
7) $(a + 4) * \frac{a}{2a + 8}$;
8) $\frac{x - 9}{4x + 8} * \frac{x^2 + 2x}{x - 9}$;
9) $\frac{4a^2 - 4a + 1}{3a + 3} * \frac{a + 1}{2a - 1}$;
10) $\frac{a^2 - 25}{4a} * \frac{4a^2}{a^2 - 5a}$.

Решение:

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

2) $\frac{2mn + n^2}{6m} * \frac{2m}{n} = \frac{n(2m + n)}{6m} * \frac{2m}{n} = \frac{2m + n}{3} * \frac{1}{1} = \frac{2m + n}{3}$

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

4) $\frac{32a}{a^2 - 9} * \frac{a - 3}{8a} = \frac{32a}{(a - 3)(a + 3)} * \frac{a - 3}{8a} = \frac{4}{a + 3} * \frac{1}{1} = \frac{4}{a + 3}$

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

6) $\frac{m - 2}{m^2 - 49} * \frac{m + 7}{m - 2} = \frac{m - 2}{(m - 7)(m + 7)} * \frac{m + 7}{m - 2} = \frac{1}{m - 7} * \frac{1}{1} = \frac{1}{m - 7}$

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

8) $\frac{x - 9}{4x + 8} * \frac{x^2 + 2x}{x - 9} = \frac{x - 9}{4(x + 2)} * \frac{x(x + 2)}{x - 9} = \frac{1}{4} * \frac{x}{1} = \frac{x}{4}$

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

10) $\frac{a^2 - 25}{4a} * \frac{4a^2}{a^2 - 5a} = \frac{(a - 5)(a + 5)}{4a} * \frac{4a^2}{a(a - 5)} = \frac{(a - 5)(a + 5)}{4a} * \frac{4a}{a - 5} = \frac{a + 5}{1} * \frac{1}{1} = a + 5$

147. Выполните умножение:
1) $\frac{3a + b}{4c} * \frac{c}{3a + b}$;
2) $\frac{ab - b^2}{8} * \frac{4a}{b^4}$;
3) $\frac{5x - 5y}{x^6} * \frac{x^3}{x - y}$;
4) $\frac{18b}{b^2 - 16} * \frac{b + 4}{3b}$;
5) $\frac{6}{m^2 - 9n^2} * (m - 3n)$;
6) $\frac{3c - 9}{9c^2 + 6c + 1} * \frac{3c + 1}{c - 3}$.

Решение:

1) $\frac{3a + b}{4c} * \frac{c}{3a + b} = \frac{1}{4} * \frac{1}{1} = \frac{1}{4}$

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

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

4) $\frac{18b}{b^2 - 16} * \frac{b + 4}{3b} = \frac{18b}{(b - 4)(b + 4)} * \frac{b + 4}{3b} = \frac{6}{b - 4} * \frac{1}{1} = \frac{6}{b - 4}$

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

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

148. Какому из данных выражений равно частное $\frac{3}{c^3} : \frac{12}{c^9}$?
1) $\frac{c^3}{4}$;
2) $\frac{c^6}{4}$;
3) $4c^3$;
4) $4c^6$.

Решение:

$\frac{3}{c^3} : \frac{12}{c^9} = \frac{3}{c^3} * \frac{c^9}{12} = \frac{1}{1} * \frac{c^6}{4} = \frac{c^6}{4}$
Ответ: 2) $\frac{c^6}{4}$

149. Выполните деление:
1) $\frac{8m}{n} : \frac{4m}{n}$;
2) $\frac{3b}{8} : b$;
3) $\frac{7c^2}{d} : \frac{c}{d^3}$;
4) $\frac{6a}{5b} : \frac{3a^2}{20b^2}$;
5) $-\frac{9a}{b^5} : \frac{18a^4}{b^3}$;
6) $a^2 : \frac{a}{b^2c}$;
7) $24a^3 : \frac{12a^2}{b}$;
8) $\frac{36a}{c^3} : (4a^2c)$.

Решение:

1) $\frac{8m}{n} : \frac{4m}{n} = \frac{8m}{n} * \frac{n}{4m} = \frac{2}{1} * \frac{1}{1} = 2$

2) $\frac{3b}{8} : b = \frac{3b}{8} * \frac{1}{b} = \frac{3}{8} * \frac{1}{1} = \frac{3}{8}$

3) $\frac{7c^2}{d} : \frac{c}{d^3} = \frac{7c^2}{d} * \frac{d^3}{c} = \frac{7c}{1} * \frac{d^2}{1} = 7cd^2$

4) $\frac{6a}{5b} : \frac{3a^2}{20b^2} = \frac{6a}{5b} * \frac{20b^2}{3a^2} = \frac{2}{1} * \frac{4b}{a} = \frac{8b}{a}$

5) $-\frac{9a}{b^5} : \frac{18a^4}{b^3} = -\frac{9a}{b^5} * \frac{b^3}{18a^4} = -\frac{1}{b^2} * \frac{1}{2a^3} = -\frac{1}{2a^3b^2}$

6) $a^2 : \frac{a}{b^2c} = a^2 * \frac{b^2c}{a} = a * \frac{b^2c}{1} = ab^2c$

7) $24a^3 : \frac{12a^2}{b} = 24a^3 * \frac{b}{12a^2} = 2a * \frac{b}{1} = 2ab$

8) $\frac{36a}{c^3} : (4a^2c) = \frac{36a}{c^3} * \frac{1}{4a^2c} = \frac{9}{c^3} * \frac{1}{ac} = \frac{9}{ac^4}$