177. Упростите выражение: 1) $\left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)$; 2) $\left(\frac{a}{b}+\frac{a+b}{a-b}\right)\cdot\frac{ab^2}{a^2+b^2}$; 3) $\left(\frac{m}{m-1}-1\right):\frac{m}{mn-n^2}$; 4) $\left(\frac{a}{b}-\frac{b}{a}\right)\cdot\frac{4ab}{a-b}$; 5) $\frac{a}{b}-\frac{a^2-b^2}{b^2}:\frac{a+b}{b}$; 6) $\frac{7x}{x+2}-\frac{x-8}{3x+6}\cdot\frac{84}{x^2-8x}$; 7) $\left(a-\frac{9a-9}{a+3}\right):\frac{a^2-3a}{a+3}$; 8) $\left(\frac{a}{a+2}-\frac{8}{a+8}\right)\cdot\frac{a^2+8a}{a-4}$.

Question

177. Упростите выражение: 1) $\left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)$; 2) $\left(\frac{a}{b}+\frac{a+b}{a-b}\right)\cdot\frac{ab^2}{a^2+b^2}$; 3) $\left(\frac{m}{m-1}-1\right):\frac{m}{mn-n^2}$; 4) $\left(\frac{a}{b}-\frac{b}{a}\right)\cdot\frac{4ab}{a-b}$; 5) $\frac{a}{b}-\frac{a^2-b^2}{b^2}:\frac{a+b}{b}$; 6) $\frac{7x}{x+2}-\frac{x-8}{3x+6}\cdot\frac{84}{x^2-8x}$; 7) $\left(a-\frac{9a-9}{a+3}\right):\frac{a^2-3a}{a+3}$; 8) $\left(\frac{a}{a+2}-\frac{8}{a+8}\right)\cdot\frac{a^2+8a}{a-4}$.

$Фото задания: 177. Упростите выражение: 1) $\left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)$; 2)…$ Фото задания

Accepted Answer

$\left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)=\frac{x(y+1)}{y}:\frac{x(y-1)}{y}=\frac{y+1}{y-1}$.
$\left(\frac{a}{b}+\frac{a+b}{a-b}\right)\frac{ab^2}{a^2+b^2}=\frac{a(a-b)+b(a+b)}{b(a-b)}\cdot\frac{ab^2}{a^2+b^2}=\frac{a^2+b^2}{b(a-b)}\cdot\frac{ab^2}{a^2+b^2}=\frac{ab}{a-b}$.
$\left(\frac{m}{m-1}-1\right):\frac{m}{mn-n^2}=\frac{1}{m-1}\cdot\frac{n(m-n)}{m}=\frac{n(m-n)}{m(m-1)}$.
$\left(\frac{a}{b}-\frac{b}{a}\right)\frac{4ab}{a-b}=\frac{a^2-b^2}{ab}\cdot\frac{4ab}{a-b}=4(a+b)$.
$\frac{a}{b}-\frac{a^2-b^2}{b^2}:\frac{a+b}{b}=\frac{a}{b}-\frac{(a-b)(a+b)}{b^2}\cdot\frac{b}{a+b}=\frac{a}{b}-\frac{a-b}{b}=1$.
$\frac{7x}{x+2}-\frac{x-8}{3(x+2)}\cdot\frac{84}{x(x-8)}=\frac{7x}{x+2}-\frac{28}{x(x+2)}=\frac{7(x^2-4)}{x(x+2)}=\frac{7(x-2)}{x}$.
$\left(a-\frac{9a-9}{a+3}\right):\frac{a^2-3a}{a+3}=\frac{a(a+3)-9(a-1)}{a+3}\cdot\frac{a+3}{a(a-3)}=\frac{a^2-6a+9}{a(a-3)}=\frac{(a-3)^2}{a(a-3)}=\frac{a-3}{a}$.
$\left(\frac{a}{a+2}-\frac{8}{a+8}\right)\frac{a^2+8a}{a-4}=\frac{a(a+8)-8(a+2)}{(a+2)(a+8)}\cdot\frac{a(a+8)}{a-4}=\frac{a(a-4)}{(a+2)(a+8)}\cdot\frac{a(a+8)}{a-4}=\frac{a^2}{a+2}$.

Ответ: 1) $\frac{y+1}{y-1}$; 2) $\frac{ab}{a-b}$; 3) $\frac{n(m-n)}{m(m-1)}$; 4) $4(a+b)$; 5) $1$; 6) $\frac{7(x-2)}{x}$; 7) $\frac{a-3}{a}$; 8) $\frac{a^2}{a+2}$.