Ответы к странице 15

32. Упростите выражение:
1) $\frac{-a}{-b}$;
2) $-\frac{-a}{b}$;
3) $-\frac{a}{-b}$;
4) $-\frac{-a}{-b}$.

Решение:

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

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

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

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

33. Восстановите равенства:
1) $\frac{a}{3} = \frac{}{6a} = \frac{}{9a^3} = \frac{}{15b} = \frac{4a^2c^3}{}$;
2) $\frac{m}{n} = \frac{4m}{} = \frac{}{2n^2} = \frac{}{mnp} = \frac{3m^4n^3}{}$.

Решение:

1) $\frac{a}{3} = \frac{}{6a} = \frac{}{9a^3} = \frac{}{15b} = \frac{4a^2c^3}{}$
$\frac{6a}{3} = 2a$, тогда:
$\frac{2a * a}{2a * 3} = \frac{2a^2}{6a}$
$\frac{9a^3}{6a} = \frac{3a^2}{2}$, тогда:
$\frac{\frac{3a^2}{2} * 2a^2}{\frac{3a^2}{2} * 6a} = \frac{3a^4}{9a^3}$
$\frac{15b}{9a^3} = \frac{5b}{3a^3}$, тогда:
$\frac{\frac{5b}{3a^3} * 3a^4}{\frac{5b}{3a^3} * 9a^3} = \frac{5ab}{15b}$
$\frac{4a^2c^3}{5ab} = \frac{4ac^3}{5b}$, тогда:
$\frac{\frac{4ac^3}{5b} * 5ab}{\frac{4ac^3}{5b} * 15b} = \frac{4a^2c^3}{12ac^3}$
Ответ:
$\frac{a}{3} = \frac{2a^2}{6a} = \frac{3a^4}{9a^3} = \frac{5ab}{15b} = \frac{4a^2c^3}{12ac^3}$

2) $\frac{m}{n} = \frac{4m}{} = \frac{}{2n^2} = \frac{}{mnp} = \frac{3m^4n^3}{}$
$\frac{4m}{m} = 4$, тогда:
$\frac{4 * m}{4 * n} = \frac{4m}{4n}$
$\frac{2n^2}{4n} = \frac{n}{2}$, тогда:
$\frac{\frac{n}{2} * 4m}{ \frac{n}{2} * 4n} = \frac{2mn}{2n^2}$
$\frac{mnp}{2n^2} = \frac{mp}{2n}$, тогда:
$\frac{\frac{mp}{2n} * 2mn}{\frac{mp}{2n} * 2n^2} = \frac{m^2p}{mnp}$
$\frac{3m^4n^3}{m^2p} = \frac{3m^2n^3}{p}$, тогда:
$\frac{\frac{3m^2n^3}{p} * m^2p}{\frac{3m^2n^3}{p} * mnp} = \frac{3m^4n^3}{3m^3n^4}$
Ответ:
$\frac{m}{n} = \frac{4m}{4n} = \frac{2mn}{2n^2} = \frac{m^2p}{mnp} = \frac{3m^4n^3}{3m^3n^4}$

34. Приведите дробь:
1) $\frac{a}{b^3}$ к знаменателю $b^5$;
2) $\frac{m}{9n}$ к знаменателю $27n^4$;
3) $\frac{6}{7x^2y}$ к знаменателю $35x^3y^2$;
4) $\frac{5k}{6p^5}$ к знаменателю $24p^9c$.

Решение:

1) $\frac{b^5}{b^3} = b^2$, тогда:
$\frac{a}{b^3} = \frac{b^2 * a}{b^2 * b^3} = \frac{ab^2}{b^5}$

2) $\frac{27n^4}{9n} = 3n^3$, тогда:
$\frac{m}{9n} = \frac{3n^3 * m}{3n^3 * 9n} = \frac{3mn^3}{27n^4}$

3) $\frac{35x^3y^2}{7x^2y} = 5xy$, тогда:
$\frac{6}{7x^2y} = \frac{5xy * 6}{5xy * 7x^2y} = \frac{30xy}{35x^3y^2}$

4) $\frac{24p^9c}{6p^5} = 4p^4c$, тогда:
$\frac{5k}{6p^5} = \frac{4p^4c * 5k}{4p^4c * 6p^5} = \frac{20kp^4c}{24p^9c}$

35. Приведите дробь:
1) $\frac{x}{y^2}$ к знаменателю $y^8$;
2) $\frac{a}{3b}$ к знаменателю $6b^3$;
3) $\frac{9}{4m^2n}$ к знаменателю $12m^3n^2$;
4) $\frac{11c}{15d^6}$ к знаменателю $30bd^7$.

Решение:

1) $\frac{y^8}{y^2} = y^6$, тогда:
$\frac{x}{y^2} = \frac{y^6 * x}{y^6 * y^2} = \frac{xy^6}{y^8}$

2) $\frac{6b^3}{3b} = 2b^2$, тогда:
$\frac{a}{3b} = \frac{2b^2 * a}{2b^2 * 3b} = \frac{2ab^2}{6b^3}$

3) $\frac{12m^3n^2}{4m^2n} = 3mn$, тогда:
$\frac{9}{4m^2n} = \frac{3mn * 9}{3mn * 4m^2n} = \frac{27mn}{12m^3n^2}$

4) $\frac{30bd^7}{15d^6} = 2bd$, тогда:
$\frac{11c}{15d^6} = \frac{2bd * 11c}{2bd * 15d^6} = \frac{22bcd}{30bd^7}$

36. Сократите дробь:
1) $\frac{a(x + 2)}{b(x + 2)}$;
2) $\frac{4(a - 6)^2}{(a - 6)^3}$;
3) $\frac{c^3(c - 4)^5}{c^6(c - 4)^3}$;
4) $\frac{2a + 2b}{7(a + b)}$;
5) $\frac{7x - 21y}{5x - 15y}$;
6) $\frac{4a - 20b}{12ab}$;
7) $\frac{6x + 12}{6x}$;
8) $\frac{a - 5b}{a^2 - 5ab}$;
9) $\frac{y^2 - 25}{10 + 2y}$;
10) $\frac{a^2 + 4a + 4}{9a + 18}$;
11) $\frac{c^2 - 6c + 9}{c^2 - 9}$;
12) $\frac{m^3 + 1}{m^2 - m + 1}$.

Решение:

1) $\frac{a(x + 2)}{b(x + 2)} = \frac{a}{b}$

2) $\frac{4(a - 6)^2}{(a - 6)^3} = \frac{4}{a - 6}$

3) $\frac{c^3(c - 4)^5}{c^6(c - 4)^3} = \frac{(c - 4)^2}{c^3}$

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

5) $\frac{7x - 21y}{5x - 15y} = \frac{7(x - 3y)}{5(x - 3y)} = \frac{7}{5}$

6) $\frac{4a - 20b}{12ab} = \frac{4(a - 5b)}{12ab} = \frac{a - 5b}{3ab}$

7) $\frac{6x + 12}{6x} = \frac{6(x + 2)}{6x} = \frac{x + 2}{x}$

8) $\frac{a - 5b}{a^2 - 5ab} = \frac{a - 5b}{a(a - 5b)} = \frac{1}{a}$

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

10) $\frac{a^2 + 4a + 4}{9a + 18} = \frac{(a + 2)^2}{9(a + 2)} = \frac{a + 2}{9}$

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

12) $\frac{m^3 + 1}{m^2 - m + 1} = \frac{m^3 + 1}{m^2 - m + 1} = \frac{(m + 1)(m^2 - m + 1)}{m^2 - m + 1} = m + 1$

37. Сократите дробь:
1) $\frac{a - b}{2(b - a)}$;
2) $\frac{3x - 6y}{4y - 2x}$;
3) $\frac{m^2 - 5mn}{15n - 3m}$;
4) $\frac{7a^4 - a^3b}{b^4 - 7ab^3}$;
5) $\frac{x^2 - 25}{5x^2 - x^3}$;
6) $\frac{y^2 - 12y + 36}{36 - y^2}$.

Решение:

1) $\frac{a - b}{2(b - a)} = -\frac{a - b}{2(a - b)} = -\frac{1}{2}$

2) $\frac{3x - 6y}{4y - 2x} = -\frac{3x - 6y}{2x - 4y} = -\frac{3(x - 2y)}{2(x - 2y)} = -\frac{3}{2}$

3) $\frac{m^2 - 5mn}{15n - 3m} = -\frac{m^2 - 5mn}{3m - 15n} = -\frac{m(m - 5n)}{3(m - 5n)} = -\frac{m}{3}$

4) $\frac{7a^4 - a^3b}{b^4 - 7ab^3} = -\frac{7a^4 - a^3b}{7ab^3 - b^4} = -\frac{a^3(7a - b)}{b^3(7a - b)} = -\frac{a^3}{b^3}$

5) $\frac{x^2 - 25}{5x^2 - x^3} = -\frac{x^2 - 25}{x^3 - 5x^2} = -\frac{(x - 5)(x + 5)}{x^2(x - 5)} = -\frac{x + 5}{x^2}$

6) $\frac{y^2 - 12y + 36}{36 - y^2} = -\frac{y^2 - 12y + 36}{y^2 - 36} = -\frac{(y - 6)^2}{(y - 6)(y + 6)} = -\frac{y - 6}{y + 6}$

38. Сократите дробь:
1) $\frac{3m - 3n}{7m - 7n}$;
2) $\frac{5a + 25b}{2a^2 + 10ab}$;
3) $\frac{4x - 16y}{16y}$;
4) $\frac{x^2 - 49}{6x + 42}$;
5) $\frac{12a^2 - 6a}{3 - 6a}$;
6) $\frac{9b^2 - 1}{9b^2 + 6b + 1}$;
7) $\frac{b^5 - b^4}{b^5 - b^6}$;
8) $\frac{7m^2 + 7m + 7}{m^3 - 1}$;
9) $\frac{64 - x^2}{3x^2 - 24x}$.

Решение:

1) $\frac{3m - 3n}{7m - 7n} = \frac{3(m - n)}{7(m - n)} = \frac{3}{7}$

2) $\frac{5a + 25b}{2a^2 + 10ab} = \frac{5(a + 5b)}{2a(a + 5b)} = \frac{5}{2a}$

3) $\frac{4x - 16y}{16y} = \frac{4(x - 4y)}{16y} = \frac{x - 4y}{4y}$

4) $\frac{x^2 - 49}{6x + 42} = \frac{(x - 7)(x + 7)}{6(x + 7)} = \frac{x - 7}{6}$

5) $\frac{12a^2 - 6a}{3 - 6a} = \frac{6a(2a - 1)}{3(1 - 2a)} = -\frac{6a(2a - 1)}{3(2a - 1)} = -\frac{6a}{3} = -2a$

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

7) $\frac{b^5 - b^4}{b^5 - b^6} = \frac{b^4(b - 1)}{b^5(1 - b)} = -\frac{b^4(b - 1)}{b^5(b - 1)} = -\frac{b^4}{b^5} = -\frac{1}{b}$

8) $\frac{7m^2 + 7m + 7}{m^3 - 1} = \frac{7(m^2 + m + 1)}{(m - 1)(m^2 + m + 1)} = \frac{7}{m - 1}$

9) $\frac{64 - x^2}{3x^2 - 24x} = \frac{(8 - x)(8 + x)}{3x(x - 8)} = -\frac{(8 - x)(8 + x)}{3x(8 - x)} = -\frac{8 + x}{3x}$