Square Root Question

The question is to simplify

sqrt(3-sqrt5) + sqrt(4+sqrt7) + sqrt(6-sqrt35)


The solution that is given is

sqrt(3-sqrt5) = sqrt(5/2) - sqrt(1/2),
sqrt(4+sqrt7) = sqrt(7/2) + sqrt(1/2),
sqrt(6-sqrt35) = sqrt(7/2) - sqrt (5/2),
and their sum is
2sqrt(7/2) = sqrt(14)


I understand that this is based on the rule

sqr(a+b*sqrt(c)) = sqrt((a+m)/2) + sqrt((a-m)/2)


This makes the question very simple.



I was trying to get to the same answer the long way by simplifying the expression without using the rule.

I rationalized the numerators, combined the first two terms with a common denominator, factor that denominator by grouping, and ended up with

2sqrt(3+sqrt5)+3sqrt(4-sqrt7)
------------------------------- + sqrt(6-sqrt35)
(4-sqrt7)(3+sqrt5)







I tried to make a common denominator for all 3, but it got very messy and there seemed like no way to group anything or find something common to eliminate anything.
I wonder if any of you math brains could tell me if I'm on the right track, going in the wrong direction, give me a hint, or just tell me it's not possible to get to sqrt(14) this way. Thanks in advance.