Ответы к странице 38

150. Найдите частное:
1) $\frac{7}{a^2} : \frac{28}{a^8}$;
2) $\frac{b^9}{8} : \frac{b^3}{48}$;
3) $\frac{27}{m^6} : \frac{36}{m^7n^2}$;
4) $\frac{6x^{10}}{y^8} : (30x^5y^2)$;
5) $49m^4 : \frac{21m}{n^2}$;
6) $\frac{16x^3y^8}{33z^5} : (-\frac{10x^2}{55z^6})$.

Решение:

1) $\frac{7}{a^2} : \frac{28}{a^8} = \frac{7}{a^2} * \frac{a^8}{28} = \frac{1}{1} * \frac{a^6}{4} = \frac{a^6}{4}$

2) $\frac{b^9}{8} : \frac{b^3}{48} = \frac{b^9}{8} * \frac{48}{b^3} = \frac{b^6}{1} * \frac{6}{1} = 6b^6$

3) $\frac{27}{m^6} : \frac{36}{m^7n^2} = \frac{27}{m^6} * \frac{m^7n^2}{36} = \frac{3}{1} * \frac{mn^2}{4} = \frac{3mn^2}{4}$

4) $\frac{6x^{10}}{y^8} : (30x^5y^2) = \frac{6x^{10}}{y^8} * \frac{1}{30x^5y^2} = \frac{x^{5}}{y^8} * \frac{1}{5y^2} = \frac{x^{5}}{5y^{10}}$

5) $49m^4 : \frac{21m}{n^2} = 49m^4 * \frac{n^2}{21m} = 7m^3 * \frac{n^2}{3} = \frac{7m^3n^2}{3}$

6) $\frac{16x^3y^8}{33z^5} : (-\frac{10x^2}{55z^6}) = \frac{16x^3y^8}{33z^5} * (-\frac{55z^6}{10x^2}) = \frac{8xy^8}{3} * (-\frac{5z}{5}) = \frac{8xy^8}{3} * (-\frac{z}{1}) = -\frac{8xy^8z}{3}$

151. Упростите выражение:
1) $\frac{a - b}{7a} : \frac{a - b}{7b}$;
2) $\frac{x^2 - y^2}{x^2} : \frac{6x + 6y}{x^5}$;
3) $\frac{c - 5}{c^2 - 4c} : \frac{c - 5}{5c - 20}$;
4) $\frac{x - y}{xy} : \frac{x^2 - y^2}{3xy}$;
5) $\frac{a^2 - 25}{a + 7} : \frac{a - 5}{a + 7}$;
6) $\frac{a^2 - 4a + 4}{a + 2} : (a - 2)$;
7) $(p^2 - 16k^2) : \frac{p + 4k}{p}$;
8) $\frac{a^2 - ab}{a^2} : \frac{a^2 - 2ab + b^2}{ab}$.

Решение:

1) $\frac{a - b}{7a} : \frac{a - b}{7b} = \frac{a - b}{7a} * \frac{7b}{a - b} = \frac{1}{a} * \frac{b}{1} = \frac{b}{a}$

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

3) $\frac{c - 5}{c^2 - 4c} : \frac{c - 5}{5c - 20} = \frac{c - 5}{c(c - 4)} : \frac{c - 5}{5(c - 4)} = \frac{c - 5}{c(c - 4)} * \frac{5(c - 4)}{c - 5} = \frac{1}{c} * \frac{5}{1} = \frac{5}{c}$

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

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

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

7) $(p^2 - 16k^2) : \frac{p + 4k}{p} = (p - 4k)(p + 4k) : \frac{p + 4k}{p} = (p - 4k)(p + 4k) * \frac{p}{p + 4k} = (p - 4k) * \frac{p}{1} = p(p - 4k)$

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

152. Выполните деление:
1) $\frac{5m - 2n}{10k} : \frac{5m - 2n}{10k^2}$;
2) $\frac{p + 3}{p^2 - 2p} : \frac{p + 3}{4p - 8}$;
3) $\frac{a^2 - b^2}{2ab} : \frac{a + b}{ab}$;
4) $\frac{a^2 - 16}{a - 3} : \frac{a + 4}{a - 3}$;
5) $\frac{y - 9}{y - 8} : \frac{y^2 - 81}{y^2 - 16y + 64}$;
6) $(x^2 - 49y^2) : \frac{x - 7y}{x}$.

Решение:

1) $\frac{5m - 2n}{10k} : \frac{5m - 2n}{10k^2} = \frac{5m - 2n}{10k} * \frac{10k^2}{5m - 2n} = \frac{1}{1} * \frac{k}{1} = k$

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

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

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

5) $\frac{y - 9}{y - 8} : \frac{y^2 - 81}{y^2 - 16y + 64} = \frac{y - 9}{y - 8} : \frac{(y - 9)(y + 9)}{(y - 8)^2} = \frac{y - 9}{y - 8} * \frac{(y - 8)^2}{(y - 9)(y + 9)} = \frac{1}{1} * \frac{y - 8}{y + 9} = \frac{y - 8}{y + 9}$

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

153. Выполните возведение в степень:
1) $(\frac{a}{b})^9$;
2) $(\frac{m}{n^2})^8$;
3) $(\frac{c}{2d})^5$;
4) $(\frac{5a^6}{b^5})^2$;
5) $(-\frac{3m^4}{2n^3})^3$;
6) $(-\frac{6a^6}{b^7})^2$.

Решение:

1) $(\frac{a}{b})^9 = \frac{a^9}{b^9}$

2) $(\frac{m}{n^2})^8 = \frac{m^8}{n^{2 * 8}} = \frac{m^8}{n^{16}}$

3) $(\frac{c}{2d})^5 = \frac{c^5}{(2d)^5} = \frac{c^5}{32d^5}$

4) $(\frac{5a^6}{b^5})^2 = \frac{(5a^6)^2}{(b^5)^2} = \frac{5^2a^{6 * 2}}{b^{5 * 2}} = \frac{25a^{12}}{b^{10}}$

5) $(-\frac{3m^4}{2n^3})^3 = -\frac{(3m^4)^3}{(2n^3)^3} = -\frac{3^3m^{4 * 3}}{2^3n^{3 * 3}} = -\frac{27m^{12}}{8n^{9}}$

6) $(-\frac{6a^6}{b^7})^2 = \frac{(6a^6)^2}{b^{7 * 2}} = \frac{6^2a^{6 * 2}}{b^{14}} = \frac{36a^{12}}{b^{14}}$

154. Представьте в виде дроби выражение:
1) $(\frac{a^6}{b^3})^{10}$;
2) $(-\frac{4m}{9n^3})^2$;
3) $(-\frac{10c^7}{3d^5})^3$;
4) $(\frac{2m^3n^2}{kp^8})^6$.

Решение:

1) $(\frac{a^6}{b^3})^{10} = \frac{a^{6 * 10}}{b^{3 * 10}} = \frac{a^{60}}{b^{30}}$

2) $(-\frac{4m}{9n^3})^2 = \frac{4^2m^2}{9^2n^{3 * 2}} = \frac{16m^2}{81n^{6}}$

3) $(-\frac{10c^7}{3d^5})^3 = -\frac{10^3c^{7 * 3}}{3^3d^{5 * 3}} = -\frac{1000c^{21}}{27d^{15}}$

4) $(\frac{2m^3n^2}{kp^8})^6 = \frac{2^6m^{3 * 6}n^{2 * 6}}{k^6p^{8 * 6}} = \frac{64m^{18}n^{12}}{k^6p^{48}}$

155. Упростите выражение:
1) $\frac{6a^4b^2}{35c^3} * \frac{14b^2}{a^7c^5} * \frac{5a^3c^8}{18b^4}$;
2) $\frac{33m^8}{34n^8} : \frac{88m^4}{51n^4} : \frac{21m^6}{16n^2}$;
3) $\frac{36x^6}{49y^5} : \frac{24x^9}{25y^4} * \frac{7x^2}{30y}$;
4) $(\frac{m^5n}{3p^3})^3 : \frac{m^{10}n^5}{54p^8}$;
5) $(\frac{2a^5}{y^6})^4 : (\frac{4a^6}{y^8})^3$;
6) $(-\frac{27x^3}{16y^5})^2 * (\frac{8y^3}{9x^2})^3$.

Решение:

1) $\frac{6a^4b^2}{35c^3} * \frac{14b^2}{a^7c^5} * \frac{5a^3c^8}{18b^4} = \frac{1}{1} * \frac{2}{1} * \frac{1}{3} = \frac{2}{3}$

2) $\frac{33m^8}{34n^8} : \frac{88m^4}{51n^4} : \frac{21m^6}{16n^2} = \frac{33m^8}{34n^8} * \frac{51n^4}{88m^4} * \frac{16n^2}{21m^6} = \frac{1}{n^2} * \frac{3}{1} * \frac{1}{7m^2} = \frac{3}{7m^2n^2}$

3) $\frac{36x^6}{49y^5} : \frac{24x^9}{25y^4} * \frac{7x^2}{30y} = \frac{36x^6}{49y^5} * \frac{25y^4}{24x^9} * \frac{7x^2}{30y} = \frac{1}{7y} * \frac{5}{2x} * \frac{1}{2y} = \frac{5}{28xy^2}$

4) $(\frac{m^5n}{3p^3})^3 : \frac{m^{10}n^5}{54p^8} = (\frac{m^5n}{3p^3})^3 * \frac{54p^8}{m^{10}n^5} = \frac{m^{15}n^3}{27p^9} * \frac{54p^8}{m^{10}n^5} = \frac{m^{5}}{p} * \frac{2}{n^2} = \frac{2m^{5}}{n^2p}$

5) $(\frac{2a^5}{y^6})^4 : (\frac{4a^6}{y^8})^3 = \frac{16a^{20}}{y^{24}} : \frac{64a^{18}}{y^{24}} = \frac{16a^{20}}{y^{24}} * \frac{y^{24}}{64a^{18}} = \frac{a^{2}}{1} * \frac{1}{4} = \frac{a^{2}}{4}$

6) $(-\frac{27x^3}{16y^5})^2 * (\frac{8y^3}{9x^2})^3 = \frac{(3^3)^2x^6}{(2^4)^2y^{10}} * \frac{(2^3)^3y^9}{(3^2)^3x^6} = \frac{3^6x^6}{2^8y^{10}} * \frac{2^9y^9}{3^6x^6} = \frac{1}{y} * \frac{2}{1} = \frac{2}{y}$