اكتب كل مقدار من المقادير التالية بأبسط صورة:

$\frac{y-5}{y+1}+\frac{y-1}{y+5}-\frac{25}{y^2+6y+5}$

$\begin{array}{rl}& \frac{y-5}{y+1}+\frac{y-1}{y+5}-\frac{25}{y^2+6y+5}=\frac{y-5}{y+1}+\frac{y-1}{y+5}-\frac{25}{(y+1)(y+5)}\\ & =\frac{(y-5)(y+5)+(y-1)(y+1)-25}{(y+1)(y+5)}=\frac{y^2-25+y^2-1-25}{(y+1)(y+5)}=\frac{2y^2-51}{(y+1)(y+5)}\end{array}$

تدرب وحل التمرينات

(1)- اكتب كل مقدار من المقادير التالية بأبسط صورة:

$\frac{x+5}{12x}\times \frac{6x-30}{x^2-25}$

$\frac{x+5}{12x}\times \frac{6x-30}{x^2-25}=\frac{x+5}{12x}\times \frac{6(x-5)}{(x-5)(x+5)}=\frac{1}{2x}$

$\frac{3-x}{4-2x}\times \frac{x^2+x-6}{9-x^2}$

$\frac{3-x}{4-2x}\times \frac{x^2+x-6}{9-x^2}=\frac{3-x}{-2(x-2)}\times \frac{(x+3)(x-2)}{(3-x)(3+x)}=\frac{-1}{2}$

$\frac{y^2-7y}{y^3-27}÷\frac{y^2-49}{y^2+3y+9}$

$\begin{array}{rl}\frac{y^2-7y}{y^3-27}÷\frac{y^2-49}{y^2+3y+9}& =\frac{y(y-7)}{(y-3)(y^2+3y+9)}\times \frac{y^2+3y+9}{(y-7)(y+7)}\\ & =\frac{y}{(y-3)(y+7)}\end{array}$

(2)- اكتب كل مقدار من المقادير التالية بأبسط صورة:

$\frac{5}{x^2-36}-\frac{2}{x^2-12x+36}$

$\begin{array}{rl}& \frac{5}{x^2-36}-\frac{2}{x^2-12x+36}=\frac{5}{(x-6)(x+6)}-\frac{2}{(x-6{)}^2}\\ & =\frac{5(x-6)-2(x+6)}{(x-6{)}^2(x+6)}=\frac{5x-30-2x-12}{(x-6{)}^2(x+6)}=\frac{3x-42}{(x-6{)}^2(x+6)}\end{array}$

$\frac{3}{x-2}-\frac{2}{x-2}+\frac{4+2x+x^2}{x^3-8}$

$\begin{array}{rl}\frac{3}{x-2}-\frac{2}{x-2}+\frac{4+2x+x^2}{x^3-8}& =\frac{1}{x-2}+\frac{4+2x+x^2}{(x-2)(x^2+2x+4)}\\ & =\frac{1+1}{x-2}=\frac{2}{x-2}\end{array}$