সরল কর :

$\left(\frac{1}{x}+\frac{1}{y}\right)\times \left(\frac{1}{y}-\frac{1}{x}\right)$

$=\left(\frac{y+x}{xy}\right)\times \left(\frac{x-y}{xy}\right)$

$=\frac{(x+y)(x-y)}{xy\times xy}$

$=\frac{(x^2-y^2)}{x^2{y}^2}$

প্রদত্ত রাশি= $\left(\frac{1}{1+x}+\frac{2x}{1-x^2}\right)\left(\frac{1}{x}-\frac{1}{x^2}\right)$

$=\frac{1+x}{(1+x)(1-x)}\times \frac{(x-1)}{x^2}$

$=\frac{1}{(1-x)}\times \frac{-(1-x)}{x^2}$

$=-\frac{1}{x^2}$

প্রদত্ত রাশি=$\left(1-\frac{c}{a+b}\right)\left(\frac{a}{a+b+c}-\frac{a}{a+b-c}\right)$

$=\frac{-2ca}{(a+b)(a+b+c)}$

প্রদত্ত রাশি =

$\left(\frac{1}{1+a}+\frac{a}{1-a}\right)\left(\frac{1}{1+a^2}-\frac{1}{1+a+a^2}\right)$

$=\left\{\frac{1(1-a)+a(1+a)}{(1+a)(1-a)}\right\}\left\{\frac{(1+a+a^2)-(1+a^2)}{(1+a^2)(1+a+a^2)}\right\}$

$=\left\{\frac{(1-a+a+a^2)}{(1+a)(1-a)}\right\}\left\{\frac{1+a+a^2-1-a^2)}{(1+a^2)(1+a+a^2)}\right\}$

$=\frac{(1+a^2)}{(1+a)(1-a)}\times \frac{a}{(1+a^2)(1+a+a^2)}$

$=\frac{a}{(1+a)(1-a)(1+a+a^2)}$

$=\frac{a}{(1-a^2)(1+a+a^2)}$

প্রদত্ত রাশি = $\left(\frac{x}{2x-y}+\frac{x}{2x+y}\right)\left(4+\frac{3y^2}{x^2-y^2}\right)$$=\left\{\frac{x(2x+y)+x(2x-y)}{(2x-y)(2x+y)}\right\}\left\{\frac{4(x^2-y^2)+3y^2}{x^2-y^2}\right\}$$=\left\{\frac{2x^2+xy+2x^2-xy}{(2x-y)(2x+y)}\right\}\left\{\frac{4x^2-4y^2+3y^2}{x^2-y^2}\right\}$$=\frac{4x^2}{(4x^2-y^2)}.\frac{(4x^2-y^2)}{(x^2-y^2)} \left[\because (a+b) (a-b)=a^2-b^2\right]$$=\frac{4x^2}{(x^2-y^2)}$

(Ans)

প্রদত্ত রাশি = $\left(\frac{2x+y}{x+y}-1\right)÷\left(1-\frac{y}{x+y}\right)$

$=\frac{2x+y-(x+y)}{x+y}÷\frac{x+y-y}{x+y}$

$=\frac{2x+y-x-y}{x+y}÷\frac{x}{x+y}$

$=\frac{x}{x+y}÷\frac{x}{x+y}$

$=\frac{x}{(x+y)}\times \frac{(x+y)}{x}$

$=1$

(Ans)

প্রদত্ত রাশি = $\left(\frac{a}{a+b}+\frac{b}{a-b}\right)÷\left(\frac{a}{a-b}-\frac{b}{a+b}\right)$

$=\frac{a^2-ab+ab+b^2}{(a+b)(a-b)}÷\frac{a^2+ab-ab+b^2}{(a-b)(a+b)}$

$=\frac{a^2+b^2}{(a+b)(a-b)}÷\frac{a^2+b^2}{(a-b)(a+b)}$

$=\frac{(a^2+b^2)}{(a+b)(a-b)}\times \frac{(a^2-b^2)}{(a^2+b^2)}$

$=1$

প্রদত্ত রাশি =$\left(\frac{a^2+b^2}{2ab}-1\right)÷\left(\frac{a^3-b^3}{a-b}-3ab\right)$

$=\left(\frac{a^2+b^2-2ab}{2ab}\right)÷\left(\frac{a^3-b^3-3ab(a-b)}{(a-b)}\right)$$=\frac{{(a-b)}^2}{2ab}\times \frac{(a-b)}{{(a-b)}^3}$

$=\frac{{(a-b)}^2}{2ab{(a-b)}^3}$

$=\frac{1}{2ab}$

প্রদত্ত রাশি = $\frac{{\left(x+y\right)}^2-4xy}{{\left(a+b\right)}^2-4ab}÷\frac{x^3-y^3-3xy\left(x-y\right)}{a^3-b^3-3ab\left(a-b\right)}$

$=\frac{{\left(x-y\right)}^2}{{\left(a-b\right)}^2}÷\frac{{\left(x-y\right)}^3}{{\left(a-b\right)}^3}$

$=\frac{{\left(x-y\right)}^2}{{\left(a-b\right)}^2}\times \frac{{\left(a-b\right)}^3}{{\left(x-y\right)}^3}$

$=\frac{a-b}{x-y}$

প্রদত্ত রাশি = $\left(\frac{a}{b}+\frac{b}{a}+1\right)÷\left(\frac{a^2}{b^2}+\frac{a}{b}+1\right)$

$=\left(\frac{a^2+b^2+ab}{ab}\right)÷\left(\frac{a^2+ab+b^2}{b^2}\right)$

$=\frac{(a^2+ab+b^2)}{ab}\times \frac{b^2}{(a^2+ab+b^2)}$

$=\frac{b}{a}$