Упростите выражения (39.19-39.21): 39.19. 1) 2a+n/a-n - 3n/a-n; 2) a² + b²/a-b -a; 3) m-n+n²/m+n; 4) a+b-a²+b²/a+b; 5) x-9/x-3 -3; 6) a²-a⁴+1/a²-1+1; 7) 2m+2n+4n²/2m-2n; 8) 2a-n+n²+2an/2a+n. 39.20. 1) 1/a+b - a²+b²/a³+b³; 2) 1/p-q - 3pq/p³-q³; 3) 1-a/a²-a+1 + a²/a³+1; 4) 6a³ +48 a/a³ + 64 - 3a²/a² - 4a + 16.

39.19

1. \(\frac{2a + n}{a - n} - \frac{3n}{a - n} = \frac{2a + n - 3n}{a - n} = \frac{2a - 2n}{a - n} = \frac{2(a - n)}{a - n} = 2\)

2. \(\frac{a^2 + b^2}{a - b} - a = \frac{a^2 + b^2 - a(a - b)}{a - b} = \frac{a^2 + b^2 - a^2 + ab}{a - b} = \frac{b^2 + ab}{a - b} = \frac{b(b + a)}{a - b}\)

3. \(m - n + \frac{n^2}{m + n} = \frac{(m - n)(m + n) + n^2}{m + n} = \frac{m^2 - n^2 + n^2}{m + n} = \frac{m^2}{m + n}\)

4. \(a + b - \frac{a^2 + b^2}{a + b} = \frac{(a + b)^2 - (a^2 + b^2)}{a + b} = \frac{a^2 + 2ab + b^2 - a^2 - b^2}{a + b} = \frac{2ab}{a + b}\)

5. \(x - \frac{9}{x - 3} - 3 = \frac{x(x - 3) - 9 - 3(x - 3)}{x - 3} = \frac{x^2 - 3x - 9 - 3x + 9}{x - 3} = \frac{x^2 - 6x}{x - 3} = \frac{x(x - 6)}{x - 3}\)

6. \(a^2 - \frac{a^4 + 1}{a^2 - 1} + 1 = \frac{a^2(a^2 - 1) - (a^4 + 1) + (a^2 - 1)}{a^2 - 1} = \frac{a^4 - a^2 - a^4 - 1 + a^2 - 1}{a^2 - 1} = \frac{-2}{a^2 - 1}\)

7. \(2m + 2n + \frac{4n^2}{2m - 2n} = \frac{(2m + 2n)(2m - 2n) + 4n^2}{2m - 2n} = \frac{4m^2 - 4n^2 + 4n^2}{2m - 2n} = \frac{4m^2}{2m - 2n} = \frac{2m^2}{m - n}\)

8. \(2a - n + \frac{n^2 + 2an}{2a + n} = \frac{(2a - n)(2a + n) + n^2 + 2an}{2a + n} = \frac{4a^2 - n^2 + n^2 + 2an}{2a + n} = \frac{4a^2 + 2an}{2a + n} = \frac{2a(2a + n)}{2a + n} = 2a\)

39.20

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

2. \(\frac{1}{p - q} - \frac{3pq}{p^3 - q^3} = \frac{1}{p - q} - \frac{3pq}{(p - q)(p^2 + pq + q^2)} = \frac{p^2 + pq + q^2 - 3pq}{(p - q)(p^2 + pq + q^2)} = \frac{p^2 - 2pq + q^2}{(p - q)(p^2 + pq + q^2)} = \frac{(p - q)^2}{(p - q)(p^2 + pq + q^2)} = \frac{p - q}{p^2 + pq + q^2}\)

3. \(\frac{1 - a}{a^2 - a + 1} + \frac{a^2}{a^3 + 1} = \frac{1 - a}{a^2 - a + 1} + \frac{a^2}{(a + 1)(a^2 - a + 1)} = \frac{(1 - a)(a + 1) + a^2}{(a + 1)(a^2 - a + 1)} = \frac{1 - a^2 + a^2}{(a + 1)(a^2 - a + 1)} = \frac{1}{a^3 + 1}\)

4. \(\frac{6a^3 + 48a}{a^3 + 64} - \frac{3a^2}{a^2 - 4a + 16} = \frac{6a(a^2 + 8)}{(a + 4)(a^2 - 4a + 16)} - \frac{3a^2}{a^2 - 4a + 16} = \frac{6a(a^2 + 8) - 3a^2(a + 4)}{(a + 4)(a^2 - 4a + 16)} = \frac{6a^3 + 48a - 3a^3 - 12a^2}{(a + 4)(a^2 - 4a + 16)} = \frac{3a^3 - 12a^2 + 48a}{(a + 4)(a^2 - 4a + 16)} = \frac{3a(a^2 - 4a + 16)}{(a + 4)(a^2 - 4a + 16)} = \frac{3a}{a + 4}\)