Question 866784
Let {{{A=sqrt(a+b)}}} and {{{B=sqrt(a-b)}}}
{{{(A+B)/(A-B)=((A+B)/(A-B))*((A+B)/(A+B))}}}
{{{(A+B)/(A-B)=(A^2+2AB+B^2)/(A^2-B^2)}}}
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{{{A^2=a+b}}}
{{{B^2=a-b}}}
{{{A^2-B^2=2b}}}
{{{2AB=2sqrt((a+b)(a-b))=2sqrt(a^2-b^2)}}}}
{{{A^2+2AB+B^2=a+b+2sqrt(a^2-b^2)+a-b=2a+2sqrt(a^2-b^2)=2(a+sqrt(a^2-b^2))}}}
Putting it all together,
{{{(A+B)/(A-B)=(2(a+sqrt(a^2-b^2)))/(2b)}}}
{{{(A+B)/(A-B)=(a+sqrt(a^2-b^2))/(b)}}}
{{{((sqrt(a+b))+(sqrt(a-b)))/((sqrt(a+b))-(sqrt(a-b)))=(a+sqrt(a^2-b^2))/(b)}}}