Question 144240

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{{{1-(1+xy^(-2))^(-2)}}}÷{{{1+(1+xy^(-2))^(-2)}}} 

Write both terms {{{xy^(-2)}}} as {{{x/y^2}}}

{{{1-(1+x/y^2)^(-2)}}}÷{{{1+(1+x/y^2)^(-2)}}}

Write the {{{1}}}'s inside the parentheses as {{{1/1}}}

{{{1-(1/1+x/y^2)^(-2)}}}÷{{{ 1+(1/1+x/y^2)^(-2)}}}

Get an LCD of {{{y^2}}} inside the parentheses and
multiply top and bottom of both {{{1/1}}}'s by {{{y^2}}}

{{{ 1-((1*y^2)/(1*y^2)+x/y^2)^(-2)}}}÷{{{ 1+((1*y^2)/(1*y^2)+x/y^2)^(-2)}}}

{{{1-(y^2/y^2+x/y^2)^(-2)}}}÷{{{ 1+((y^2)/(y^2)+x/y^2)^(-2)}}}

Combine the numerators over the LCD in the parentheses:

{{{ 1-((y^2+x)/y^2 )^(-2)}}}÷{{{ 1+((y^2+x)/y^2)^(-2)}}}

Now use this principle on those fractions to negative powers:
                {{{(A/B)^(-N) = (B/A)^N}}}

{{{ 1-(y^2/(y^2+x) )^2}}}÷{{{ 1+(y^2/(y^2+x))^2}}}

Square tops and bottoms:

{{{ 1-y^4/(y^2+x)^2}}}÷{{{ 1+y^4/(y^2+x)^2}}}

Write the {{{1}}}'s as {{{1/1}}}

{{{ 1/1-y^4/(y^2+x)^2}}}÷{{{ 1/1+y^4/(y^2+x)^2}}}

Get an LCD of {{{(y^2+x)^2}}} and multiply the {{{1/1}}}'s
top and bottom by it:

{{{ (1*(y^2+x)^2)/(1*(y^2+x)^2)-y^4/(y^2+x)^2}}}÷{{{(1*(y^2+x)^2)/(1*(y^2+x)^2)+y^4/(y^2+x)^2}}}

{{{ (y^2+x)^2/(y^2+x)^2-y^4/(y^2+x)^2}}}÷{{{(y^2+x)^2/(y^2+x)^2+y^4/(y^2+x)^2}}}

Combine the numerators over the LCD

{{{ ((y^2+x)^2-y^4)/(y^2+x)^2}}}÷{{{((y^2+x)^2+y^4)/(y^2+x)^2}}}

Invert the second fraction and change division to multiplication:

{{{ ((y^2+x)^2-y^4)/(y^2+x)^2}}}×{{{(y^2+x)^2/((y^2+x)^2+y^4)}}}

Now the {{{(y^2+x)^2}}}'s cancel

{{{ ((y^2+x)^2-y^4)/cross((y^2+x)^2)}}}×{{{cross((y^2+x)^2)/((y^2+x)^2+y^4)}}}

{{{ ((y^2+x)^2-y^4)/((y^2+x)^2+y^4)}}} 

Now we square {{{ (y^2+x)^2 }}} = {{{ (y^2+x)(y^2+x) }}} = {{{ y^4+2xy^2+x^2 }}}

And replace in the numerator and denominator

{{{ ((y^4+2xy^2+x^2)-y^4)/((y^4+2xy^2+x^2)+y^4)}}}

{{{ (y^4+2xy^2+x^2-y^4)/(y^4+2xy^2+x^2+y^4)}}}

Combine terms in numerator and denominator

{{{ (2xy^2+x^2)/(2y^4+2xy^2+x^2)}}}

Edwin<pre>