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.

Use x^½ terminology instead of sqrt(x), it makes the simplifications more obvious imo.

Sorry I can't help you out, brain is stuck in lazy gear today.

C'mon guys. I know this question is trivial in comparison to our big picture discussions, but this is bugging me.
I spent an hour working on this last night, and I need to know if I got anywhere. [img]/images/graemlins/smile.gif[/img]
This must be an easy question for the math brains and assorted PhD's out there. Help out a lowly high school Math/Physics teacher.

[Trantor]

It seems to me whatever manipulation you do along the lines you are trying will always retain terms of the form sqrt(a +/- sqrt(b)). That means you won't be able to cancel out terms to get to the answer. The equation you first mentions works to simplify the collection of term because it reduces them all to the same type , ie sqrt(x), namely the type of term you are trying to end up with.

Thus it seems to me your approach can't work. Don't know if this helps but I've convinced myself there is no point in trying for the type of approach you are tryng.