The value of (√5 + √3)/(√5 − √3) + (√5 − √3)/(√5 + √3) is:

Putting the two fractions over the common denominator (√5 − √3)(√5 + √3) = 5 − 3 = 2 gives [$(\surd 5+\surd 3{)}^2+(\surd 5-\surd 3{)}^2$]/2. The cross terms ±2√15 cancel, leaving $(8+8)/2=16/2=8$. (16 forgets to divide by the denominator 2; 4 writes the numerator as 5 + 3 = 8 instead of $2(5+3)=16$; 2√15 is the cross term that cancels.) Numerically the two fractions are about 15.47 and 0.065, adding to 8.