335 Вправа 335Алгебра 8 клас Бевз 2025 · § 7 · стор. 68

335. Спрости вираз.

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

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

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

=${\displaystyle \frac{-mn}{m-n}=}$${\displaystyle \frac{mn}{n-m}}$;

3) ${\displaystyle \frac{m^2}{m-n}:\frac{mn}{n-m}=}$${\displaystyle \frac{m^2}{m-n}\cdot \frac{n-m}{mn}=}$${\displaystyle \frac{m^2\cdot (n-m)}{(m-n)\cdot mn}}$=

=${\displaystyle -\frac{\cancel{m}m\cancel{\left(m-n\right)}}{\cancel{m}n\cancel{\left(m-n\right)}}=}$${\displaystyle -\frac{m}{n}}$;

б) ${\displaystyle \frac{x-\frac{yz}{y-z}}{y-\frac{xz}{x-z}}=}$${\displaystyle (x-\frac{yz}{y-z}):(y-\frac{xz}{x-z})}$;

1) ${\displaystyle \frac{x^{\mathrm{\setminus }y-z}}{1}-\frac{yz}{y-z}=}$${\displaystyle \frac{x(y-z)-yz}{y-z}=}$${\displaystyle \frac{xy-xz-yz}{y-z}}$;

2) ${\displaystyle \frac{y^{\mathrm{\setminus }x-z}}{1}-\frac{xz}{x-z}=}$${\displaystyle \frac{y(x-z)-xz}{x-z}}$=

=${\displaystyle \frac{xy-yz-xz}{x-z}=}$${\displaystyle \frac{yx-yz-xz}{x-z}}$;

3) ${\displaystyle \frac{xy-xz-yz}{y-z}:\frac{xy-xz-yz}{x-z}}$=

=${\displaystyle \frac{xy-xz-yz}{y-z}\cdot \frac{x-z}{xy-xz-yz}}$=

=${\displaystyle \frac{\cancel{\left(xy-xz-yz\right)}(x-z)}{\cancel{\left(xy-xz-yz\right)}(y-z)}=}$${\displaystyle \frac{x-z}{y-z}}$.