Question 801201
I think you have been asked to rationalize the expression {{{sqrt(A) / (sqrt(A) + 2sqrt(B))}}}(which should have been written as sqrt(A) / (sqrt(A) + 2sqrt(B)) ,
or at least sqrtA / (sqrtA + 2sqrtB) .
The parentheses around the denominator determine the meaning of the expression).


{{{sqrt(A) / (sqrt(A) + 2sqrt(B))  }}} =  {{{sqrt(A)(sqrt(A) - 2sqrt(B))/ ((sqrt(A) + 2sqrt(B))(sqrt(A) - 2sqrt(B)))   }}} = {{{(A - 2sqrt(A)sqrt(B))/ (A  - (2sqrt(B))^2)   }}} =  {{{(A - 2sqrt(A)sqrt(B))/ (A  - 4B^2)   }}}
 
Your last expression could be written as
(sqrtA)(sqrtA - 2sqrtB) / ((sqrtA + 2sqrtB)(sqrtA - 2sqrtB)) ={{{sqrt(A)(sqrt(A) - 2sqrt(B))/ ((sqrt(A) + 2sqrt(B))(sqrt(A) - 2sqrt(B)))   }}}
but the line in between does not make sense.