390° Вправа 390Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 109

390. Спростіть вираз:

Показники однакові: ${\displaystyle A^n\cdot B^n=}$${\displaystyle {\left(AB\right)}^n}$, ${\displaystyle A^n:B^n=}$${\displaystyle {\left(\frac{A}{B}\right)}^n}$.

1) ${\displaystyle {\left(\frac{x}{y}\right)}^{-5}\cdot {\left(\frac{y}{x}\right)}^{-5}=}$${\displaystyle {\left(\frac{\cancel{x}\cdot \cancel{y}}{\cancel{y}\cdot \cancel{x}}\right)}^{-5}=}$${\displaystyle 1^{-5}=}$${\displaystyle 1}$;

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

3) ${\displaystyle {\left(\frac{a}{a+b}\right)}^{-3}\cdot {\left(\frac{a+b}{b}\right)}^{-3}=}$${\displaystyle {\left(\frac{a\cdot \cancel{\left(a+b\right)}}{\cancel{\left(a+b\right)}\cdot b}\right)}^{-3}=}$${\displaystyle {\left(\frac{a}{b}\right)}^{-3}=}$${\displaystyle \frac{b^3}{a^3}}$;

4) ${\displaystyle {(z-6)}^{-8}\cdot {\left(\frac{z}{z-6}\right)}^{-8}=}$${\displaystyle {\left(\frac{\cancel{\left(z-6\right)}\cdot z}{\cancel{z-6}}\right)}^{-8}=}$${\displaystyle z^{-8}=}$${\displaystyle \frac{1}{z^8}}$;

5) ${\displaystyle {(n+5)}^{-7}\cdot {\left(\frac{n}{n+5}\right)}^{-7}\cdot {\left(\frac{1}{n}\right)}^{-7}=}$${\displaystyle {\left(\frac{\cancel{\left(n+5\right)}\cdot \cancel{n}\cdot 1}{\cancel{\left(n+5\right)}\cdot \cancel{n}}\right)}^{-7}=}$${\displaystyle 1^{-7}=}$${\displaystyle 1}$;

6) ${\displaystyle {\left(\frac{1}{b+c}\right)}^{-1}:{\left(\frac{2}{b+c}\right)}^{-1}=}$${\displaystyle {(\frac{1}{b+c}:\frac{2}{b+c})}^{-1}=}$${\displaystyle {\left(\frac{1\cdot \cancel{\left(b+c\right)}}{\cancel{\left(b+c\right)}\cdot 2}\right)}^{-1}=}$${\displaystyle {\left(\frac{1}{2}\right)}^{-1}=}$${\displaystyle 2}$;

7) ${\displaystyle {\left(\frac{a}{bc}\right)}^{-6}:{\left(\frac{a^2}{bc}\right)}^{-6}=}$${\displaystyle {(\frac{a}{bc}:\frac{a^2}{bc})}^{-6}=}$${\displaystyle {\left(\frac{\cancel{a}\cdot \cancel{bc}}{\cancel{bc}\cdot a\cdot \cancel{a}}\right)}^{-6}=}$${\displaystyle {\left(\frac{1}{a}\right)}^{-6}=}$${\displaystyle a^6}$;

8) ${\displaystyle \frac{n^{-5}}{m^{-5}}:{\left(\frac{m^6}{2n^{-4}}\right)}^{-5}=}$${\displaystyle {\left(\frac{n}{m}\right)}^{-5}:{\left(\frac{m^6}{2n^{-4}}\right)}^{-5}=}$${\displaystyle {\left(\frac{n\cdot 2n^{-4}}{m\cdot m^6}\right)}^{-5}=}$${\displaystyle {\left(\frac{2n^{-3}}{m^7}\right)}^{-5}=}$${\displaystyle \frac{2^{-5}{n}^{15}}{m^{-35}}=}$${\displaystyle \frac{m^{35}{n}^{15}}{32}}$.