Question

упростить выражение
$frac{1}{ x^{2} } + frac{x-2}{x}$

$frac{2mn}{ m^{2} + n^{2} } + frac{2m}{ m^{2} - n^{2} } - frac{1}{m-n}$

Answer 1

$frac{1}{ x^{2} } + frac{x-2}{x} =frac{1+x(x-2)}{ x^{2} }=frac{1+x^{2} -2x}{ x^{2} }=frac{(x-1)^2}{ x^{2} }$
$frac{2mn}{ m^{2} + n^{2} } + frac{2m}{ m^{2} - n^{2} } - frac{1}{m-n} =frac{2mn}{ m^{2} + n^{2} } + frac{2m}{ (m - n)(m+n) } - frac{1}{m-n}$ = $frac{2mn}{ m^{2} + n^{2} } + frac{2m-(m+n)}{ (m - n)(m+n) }=frac{2mn}{ m^{2} + n^{2} } + frac{2m-m-n}{ (m - n)(m+n) }$ = $frac{2mn}{ m^{2} + n^{2} } + frac{m-n}{ (m - n)(m+n) }=frac{2mn}{ m^{2} + n^{2} } + frac{1}{ m+n }$ = $frac{2mn(m+n)+m^2+n^2}{(m^{2} + n^{2})(m+n)}=frac{2m^2n+2mn^2+m^2+n^2}{(m^{2} + n^{2})(m+n)}$ = $frac{2m^2n+2mn^2+m^2+n^2+m^2n-m^2n+mn^2-mn^2}{(m^{2} + n^{2})(m+n)}$ = $frac{(3m^2n+3mn^2+m^2+n^2)-m^2n-mn^2}{(m^{2} + n^{2})(m+n)} = frac{(m+n)^3-mn(m+n)}{(m^{2} + n^{2})(m+n)}$ = $frac{(m+n)((m+n)^2-mn)}{(m^{2} + n^{2})(m+n)} = frac{(m+n)^2-mn}{m^{2} + n^{2}}$ = $frac{m^2+2mn+n^2-mn}{m^{2} + n^{2}} = frac{m^2+mn+n^2}{m^{2} + n^{2}}$