1362 Вправа 1362Алгебра 8 клас Бевз 2025 · Повторення · стор. 262

1362. Спрости вираз (m³ − mn²)/(m² + n²) · (n/(m³ − m²n + mn²) + (m − 2n)/(m³ + n³)).

Розкладаємо: ${\displaystyle m^3-m{n}^2=}$${\displaystyle m(m-n)(m+n)}$;

${\displaystyle m^3-m^{2}n+m{n}^2=}$${\displaystyle m(m^2-mn+n^2)}$;

${\displaystyle m^3+n^3=}$${\displaystyle (m+n)(m^2-mn+n^2)}$.

1) ${\displaystyle \frac{n^{\mathrm{\setminus }m+n}}{m(m^2-mn+n^2)}+\frac{{m-2n}^{\mathrm{\setminus }m}}{(m+n)(m^2-mn+n^2)}}$=

=${\displaystyle \frac{mn+n^2+m^2-2mn}{m(m+n)(m^2-mn+n^2)}=}$${\displaystyle \frac{\cancel{m^2-mn+n^2}}{m(m+n)\cancel{\left(m^2-mn+n^2\right)}}=}$${\displaystyle \frac{1}{m\left(m+n\right)}}$;

2) ${\displaystyle \frac{m(m-n)(m+n)}{m^2+n^2}\cdot \frac{1}{m(m+n)}=}$${\displaystyle \frac{m-n}{m^2+n^2}}$.