409 Вправа 409Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 112

409. Спростіть вираз:

1) ${\displaystyle {\left(\frac{b^{-3}-1}{b+1}\right)}^{-2}\cdot {\left(\frac{2b+2}{b^{-2}+b^{-1}+1}\right)}^{-2}=}$

(${\displaystyle b^{-3}-1=}$${\displaystyle (b^{-1}-1)(b^{-2}+b^{-1}+1)}$)

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

2) ${\displaystyle {\left(a{\left(\frac{a}{b}\right)}^{-2}\right)}^{-1}:{\left({\left(\frac{a}{b}\right)}^{-1}b\right)}^2=}$

(${\displaystyle a\cdot {\left(\frac{a}{b}\right)}^{-2}=}$${\displaystyle \frac{b^2}{a}}$; ${\displaystyle {\left(\frac{a}{b}\right)}^{-1}\cdot b=}$${\displaystyle \frac{b^2}{a}}$)

${\displaystyle =}$${\displaystyle {\left(\frac{b^2}{a}\right)}^{-1}:{\left(\frac{b^2}{a}\right)}^2=}$${\displaystyle {\left(\frac{b^2}{a}\right)}^{-3}=}$${\displaystyle \frac{a^3}{b^6}}$;

3) ${\displaystyle ({\left(\frac{n}{m}\right)}^{-2}-{\left(\frac{m}{n}\right)}^{-2})\cdot {\left(mn\right)}^2=}$${\displaystyle (\frac{m^2}{n^2}-\frac{n^2}{m^2})\cdot m^2{n}^2=}$${\displaystyle \frac{m^2}{\cancel{n^2}}\cdot m^2\cancel{n^2}-\frac{n^2}{\cancel{m^2}}\cdot \cancel{m^2}{n}^2=}$${\displaystyle m^4-n^4}$;

4) ${\displaystyle \frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}}-\frac{x^{-1}-y^{-1}}{x^{-1}+y^{-1}}=}$

(чисельник і знаменник кожного дробу множимо на ${\displaystyle xy}$)

${\displaystyle =}$${\displaystyle \frac{{y+x}^{\mathrm{\setminus }x+y}}{y-x}-\frac{{y-x}^{\mathrm{\setminus }y-x}}{y+x}=}$${\displaystyle \frac{(y+x)\cdot \left(x+y\right)}{(y-x)(y+x)}-\frac{(y-x)\cdot \left(y-x\right)}{(y-x)(y+x)}=}$${\displaystyle \frac{\left(x+y\right)(x+y)-\left(y-x\right)(y-x)}{(y-x)(y+x)}=}$${\displaystyle \frac{\cancel{x^2}+2xy\cancel{+y^2}\cancel{-y^2}+2xy\cancel{-x^2}}{y^2-x^2}=}$${\displaystyle \frac{4xy}{y^2-x^2}}$;

5) ${\displaystyle \frac{n+n^2+n^3+n^4}{n^{-1}+n^{-2}+n^{-3}+n^{-4}}=}$

(${\displaystyle n^{-1}+n^{-2}+n^{-3}+n^{-4}=}$${\displaystyle \frac{n^3+n^2+n+1}{n^4}}$)

${\displaystyle =}$${\displaystyle \frac{n\cancel{\left(1+n+n^2+n^3\right)}\cdot n^4}{\cancel{n^3+n^2+n+1}}=}$${\displaystyle n^5}$;

6) ${\displaystyle \frac{x{y}^{-9}+x^{-9}y}{x^{-2}{y}^8+x^8{y}^{-2}}=}$

(${\displaystyle x{y}^{-9}+x^{-9}y=}$${\displaystyle \frac{x^{10}+y^{10}}{x^9{y}^9}}$; ${\displaystyle x^{-2}{y}^8+x^8{y}^{-2}=}$${\displaystyle \frac{y^{10}+x^{10}}{x^2{y}^2}}$)

${\displaystyle =}$${\displaystyle \frac{\cancel{\left(x^{10}+y^{10}\right)}\cdot \cancel{x^2{y}^2}}{x^7{y}^7\cdot \cancel{x^2{y}^2}\cdot \cancel{\left(x^{10}+y^{10}\right)}}=}$${\displaystyle \frac{1}{x^7{y}^7}}$.