+128407 sqrt [ 17 + 12sqrt (2) ]
Let us suppose that we can write this in the form a + b sqrt (2) where a, b are both positive integers
So
sqrt [ 17 + 12sqrt(2) ] = a + b sqrt (2) square both sides
17 + 12 sqrt (2) = a^2 + 2ab sqrt (2) + 2b^2
Equating coefficients it must be that
12sqrt (2) = 2ab sqrt (2) and a^2 + 2b^2 = 17 (2)
12 = 2ab
6 = ab
b = 6/a (1)
Sub (1) into (2) and we have that
a^2 + 2 (6/a)^2 =17
a^2 + 2*36 / a^2 = 17
a^2 + 72/a^2 = 17 multiply through by a^2
a^4 + 72 = 17a^2
a^4 - 17a^2 + 72 = 0 factor
(a^2 - 8) ( a^2 - 9) = 0
We want a to be a positive integer so a = 3
And b = 6/a= 6/3 =2
So.....sqrt [ 17 + 12sqrt (2) ] can be simplified to
3 + 2sqrt (2)
