410 Вправа 410Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 112

410. Спростіть вираз:

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

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

3) ${\displaystyle \frac{x^{-1}+x^{-2}+x^{-3}}{x+x^2+x^3}=}$${\displaystyle \frac{\frac{x^2+x+1}{x^3}}{x(1+x+x^2)}=}$${\displaystyle \frac{\cancel{x^2+x+1}}{x^3\cdot x\cdot \cancel{\left(x^2+x+1\right)}}=}$${\displaystyle x^{-4}}$;

4) ${\displaystyle \frac{m{n}^{-6}+m^{-2}}{m^3{n}^{-8}+n^{-2}}=}$${\displaystyle \frac{\frac{m^3+n^6}{m^2{n}^6}}{\frac{m^3+n^6}{n^8}}=}$${\displaystyle \frac{\cancel{\left(m^3+n^6\right)}\cdot n^8}{m^2{n}^6\cdot \cancel{\left(m^3+n^6\right)}}=}$${\displaystyle \frac{n^2}{m^2}}$.