Преобразуйте в алгебраическую дробь:
а) $\frac{2a}{a^2 - 9} + \frac{3}{a - 3}$;
б) $\frac{5}{m + n} - \frac{4n}{m^2 - n^2}$;
в) $\frac{x}{4 - 9x^2} + \frac{1}{3x + 2}$;
г) $\frac{1}{2p + 4q} - \frac{q}{4q^2 - p^2}$;
д) $\frac{1}{a^2 + ab + b^2} + \frac{b}{a^3 - b^3}$;
е) $\frac{m^2 + n^2}{m^3 + n^3} - \frac{1}{2(m + n)}$;
ж) $\frac{x^2 - 2xy}{(x - 2y)^3} + \frac{1}{2y - x}$;
з) $\frac{2(p + q)}{p^3 - q^3} + \frac{3}{q^2 - p^2}$.
$\frac{2a}{a^2 - 9} + \frac{3}{a - 3} = \frac{2a}{(a - 3)(a + 3)} + \frac{3}{a - 3} = \frac{2a + 3(a + 3)}{(a - 3)(a + 3)} = \frac{2a + 3a + 9}{(a - 3)(a + 3)} = \frac{5a + 9}{a^2 - 9}$
$\frac{5}{m + n} - \frac{4n}{m^2 - n^2} = \frac{5}{m + n} - \frac{4n}{(m - n)(m + n)} = \frac{5(m - n) - 4n}{(m - n)(m + n)} = \frac{5m - 5n - 4n}{(m - n)(m + n)} = \frac{5m - 9n}{m^2 - n^2}$
$\frac{x}{4 - 9x^2} + \frac{1}{3x + 2} = \frac{x}{(2 - 3x)(2 + 3x)} + \frac{1}{3x + 2} = \frac{x + 2 - 3x}{(2 - 3x)(2 + 3x)} = \frac{2 - 2x}{(2 - 3x)(2 + 3x)} = \frac{2(1 - x)}{4 - 9x^2}$
$\frac{1}{2p + 4q} - \frac{q}{4q^2 - p^2} = \frac{1}{2(p + 2q)} - \frac{q}{(2q - p)(2q + p)} = \frac{2q - p - 2q}{2(2q - p)(2q + p)} = \frac{-p}{2(2q - p)(2q + p)} = -\frac{p}{2(4q^2 - p^2)}$
$\frac{1}{a^2 + ab + b^2} + \frac{b}{a^3 - b^3} = \frac{1}{a^2 + ab + b^2} + \frac{b}{(a - b)(a^2 + ab + b^2)} = \frac{a - b + b}{(a - b)(a^2 + ab + b^2)} = \frac{a}{a^3 - b^3}$
$\frac{m^2 + n^2}{m^3 + n^3} - \frac{1}{2(m + n)} = \frac{m^2 + n^2}{(m + n)(m^2 - mn + n^2)} - \frac{1}{2(m + n)} = \frac{2(m^2 + n^2) - (m^2 - mn + n^2)}{2(m + n)(m^2 - mn + n^2)} = \frac{2m^2 + 2n^2 - m^2 + mn - n^2}{2(m + n)(m^2 - mn + n^2)} = \frac{m^2 + n^2 + mn}{2(m^3 + n^3)}$
$\frac{x^2 - 2xy}{(x - 2y)^3} + \frac{1}{2y - x} = \frac{x(x - 2y)}{(x - 2y)^3} + \frac{1}{2y - x} = \frac{x}{(x - 2y)^2} + \frac{1}{2y - x} = \frac{x}{(2y - x)^2} + \frac{1}{2y - x} = \frac{x + 2y - x}{(2y - x)^2} = \frac{2y}{(x - 2y)^2}$
$\frac{2(p + q)}{p^3 - q^3} + \frac{3}{q^2 - p^2} = \frac{2(p + q)}{(p - q)(p^2 + pq + q^2)} + \frac{3}{(q - p)(q + p)} = \frac{2(p + q)}{(p - q)(p^2 + pq + q^2)} - \frac{3}{(p - q)(p + q)} = \frac{2(p + q)(p + q) - 3(p^2 + pq + q^2)}{(p - q)(p + q)(p^2 + pq + q^2)} = \frac{2(p + q)^2 - 3p^2 - 3pq - 3q^2}{(p - q)(p + q)(p^2 + pq + q^2)} = \frac{2(p^2 + 2pq + q) - 3p^2 - 3pq - 3q^2}{(p - q)(p + q)(p^2 + pq + q^2)} = \frac{2p^2 + 4pq + 2q^2 - 3p^2 - 3pq - 3q^2}{(p - q)(p + q)(p^2 + pq + q^2)} = \frac{-p^2 + pq - q^2}{(p - q)(p + q)(p^2 + pq + q^2)} = \frac{-p^2 + pq - q^2}{(p^3 - q^3)(p + q)}$
Пожауйста, оцените решение
