M2 SURDS

[√(m + x) + √(m - x)] / [√(m + x) - √(m - x)]
= [√(m + x) + √(m - x)]^2 / {[√(m + x) - √(m - x)][√(m + x) + √(m - x)]}
= {m + x + 2√[(m + x)(m - x)] + m - x} / (m + x - m + x)
= [m + √(m^2 - x^2)] / x or (m/x) + √[(m/x)^2 - 1] ... (1)
x = 2mn / (n^2 + 1)
m/x = (n^2 + 1)/2n
(1) becomes (n^2 + 1)/2n + √{[(n^2 + 1)/2n]^2 - 1}
= (n^2 + 1)/2n + √(n^4 + 2n^2 + 1 - 4n^2) / 2n
= (n^2 + 1)/2n + √(n^4 - 2n^2 + 1) / 2n
= (n^2 + 1)/2n + √(n^2 - 1)^2 / 2n
= (n^2 + 1)/2n + (n^2 - 1) / 2n
= n