This article explains how to simplify the expression 41/2 + 31/3 / 41/2 - 31/3 using the difference of cubes formula. It first sets x = 41/2 and y = 31/3 in the formula x3 - y3 = (x - y)(x2 + xy + y2). It then multiplies the numerator and denominator of the original expression by x2 + xy + y2. This reduces the denominator to x3 - y3, allowing the final simplification of the expression to 11 + 8√3 + 4√9/5.