12/(2(2)^(1/2)(2^(1/2) +6)^2)

12 / [2(2)^(1/2)(2^(1/2) + 6)^2]

Since (2)^(1/2) = √2:

12 / [2√2(2^(1/2) + 6)^2]

Substituting again:

12 / [2√2(√2 + 6)^2]

Using FOIL on (√2 + 6)^2  =  2 + 12√2 + 36  =  12√2 + 38

Substituing:

12 /  [2√2(12√2 + 38)]

=  12 / [48 + 76√2]

Dividing top and bottom by 4:

=  3 / [12 + 19√2]

To write into reduced radical form, multiply both the numerator and the denominator by 12 - 19√2:

=  3[ 12 - 19√2 ] / [ (12 + 19√2)(12 - 19√2)]

=  (36 - 57√2) / -578  

=  (57√2 - 36) / 578