secondary school maths

Let √(√80 + 12) = √a + √b where a and b are integers
√80 + 12 = a + b + 2√(ab)
Rational part a + b = 12
a = 12 - b ... (1)
Irrational part √80 = 2√(ab)
Use (1), √80 = 2√[(12 - b)b]
80 = 4(12 - b)b
b^2 - 12b - 20 = 0
(b - 10)(b - 2) = 0
b = 2 or b = 10
when b = 2, a = 10
when b = 10, b = 2
Therefore √(√80 + 12) = √2 + √10

((80^(1/2))+12)^(1/2)=?
Sol
(80^(1/2))+12
=12+2(20)^(1/2)
=2+2(20)^(1/2)+10
=[2^(1/2)]^2+2*(2^(1/2))*(10^(1/2))+[10^(1/2)]^2
=[2^(1/2)+10^(1/2)]^2
((80^(1/2))+12)^(1/2)
=2^(1/2)+10^(1/2)