Fully simplify the expression: 4 / (sqrt(8) + 4)

Notice that the square root of 8 can be simplified: 4 / (sqrt(4) * sqrt(2) + 4) = 4 / (2 * sqrt(2) + 4) Divide top and bottom of the fraction by two: 2 / (sqrt(2) + 2) Rationalise the denominator: (2 / (sqrt(2) + 2)) * ((sqrt(2) - 2) / (sqrt(2) - 2)) = (2*sqrt(2) - 4) / ((sqrt(2) + 2) * (sqrt(2) - 2)) Expand brackets on denominator: (2 * sqrt(2) - 4) / (2 - 4) = (2 * sqrt(2) - 4) / -2 = (-sqrt(2) + 2) = 2 - sqrt(2)