Need to calculate a+b

Question

2(sqrt(3+sqrt(5-sqrt(13+sqrt(48))))= sqrt(a) +sqrt(b). a and b are natural numbers. Find a+b

Answer 1

Let's first simplify a little bit. Start from the last sq root: 13+sqrt(48))))

13+√48=12+4√3+1= (2√3)²+2 *1*2*√3+1²=(2*√3+1)²

Now

√5-(13+√48)=√5-(2*√3+1)=√4-2*√3)=√1²-2*1*√3+(√3)²=

=√(1-√3)²=(1-√3),

 

Now the last (basically the first)

2(√3+1-√3) = 2(√(4-√3))= √16-4√3= √12-4√3+4 =
√(2√3)²-2*2*√3+2²)=√(2√3-2)²=2√3-2.

 

So far what we have is

2(sqrt(3+sqrt(5-sqrt(13+sqrt(48))))=2(2√3-2)=2(√3-1)

Therefore

2(√3-1)=√a+√b (*)

Square both sides of (*)

4(4-2√3)=a+b-2√(ab)

 Since a and b are natural numbers, I think you can figure our what should be a+b.