466 Вправа 466Алгебра 8 клас Бевз 2025 · § 9 · стор. 93

466. Спрости вираз.

а) ${\displaystyle (a^{-1}-c^{-1}):\frac{a^2-c^2}{ac}}$;

${\displaystyle (\frac{1}{a}-\frac{1}{c}):\frac{a^2-c^2}{ac}=}$${\displaystyle \frac{c-a}{ac}\cdot \frac{ac}{(a-c)(a+c)}}$=

=${\displaystyle -\frac{\cancel{\left(a-c\right)}\cancel{ac}}{\cancel{ac}\cancel{\left(a-c\right)}(a+c)}=}$${\displaystyle -\frac{1}{a+c}}$;

б) ${\displaystyle \frac{1}{a^{-3}-x^{-3}}-\frac{1}{a^{-3}+x^{-3}}}$;

1) ${\displaystyle \frac{1}{a^{-3}-x^{-3}}=}$${\displaystyle 1:\frac{x^3-a^3}{a^3{x}^3}=}$${\displaystyle \frac{a^3{x}^3}{x^3-a^3}}$;

2) ${\displaystyle \frac{1}{a^{-3}+x^{-3}}=}$${\displaystyle \frac{a^3{x}^3}{x^3+a^3}}$;

3) ${\displaystyle \frac{a^3{x}^3}{x^3-a^3}-\frac{a^3{x}^3}{x^3+a^3}=}$${\displaystyle \frac{a^3{x}^3(x^3+a^3-x^3+a^3)}{x^6-a^6}=}$${\displaystyle \frac{2a^6{x}^3}{x^6-a^6}}$;

в) ${\displaystyle {(m^{-1}+n^{-1})}^2+{(m^{-1}-n^{-1})}^2}$;

${\displaystyle m^{-2}+2m^{-1}{n}^{-1}+n^{-2}+m^{-2}-2m^{-1}{n}^{-1}+n^{-2}}$=

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

г) ${\displaystyle {(c^{-2}-c^2)}^2-c^{-4}+2}$;

${\displaystyle c^{-4}-2+c^4-c^{-4}+2=}$${\displaystyle c^4}$.