a = sqrt [ 4 + sqrt (4 + a) ] square both sides
a^2 = 4 + sqrt ( 4 + a)
a^2 - 4 = sqrt (4 + a) square both sides again
a^4 - 8a^2 + 16 = 4 + a
a^4 - 8a^2 - a + 12 = 0
There are 4 possible real solutions to this
The only correct one is ... a = [ 1 + sqrt (17 ] / 2
Likewise...setting each of the other equations up as 4th power polynomials by squaring each side twice produces
b = [ 1 + sqrt (13 ) ] / 2
c = [-1 + sqrt(17) ] / 2
d = [ -1 + sqrt (13) ] / 2
So....abcd = acbd =
[ 1 + sqrt (17)] / 2 * [-1 + sqrt(17) ] /2 * [ 1 + sqrt (13 ) ] / 2 * [ -1 + sqrt (13) ] / 2 =
[ -1 + 17 ] / 4 * [ -1 + 13] / 4 =
[ 16 / 4 ] * [ 12 / 4 ] =
4 * 3 =
12
+476 
