To rationalize the denominator we need to remove all of the radicals from the denominator by multiplying by the appropriate form of 1. For this type of denominator remember the rule:
(a + b) xx (a - b) = a^2 - b^2
(2sqrt(27) + sqrt(8))/(2sqrt(27) + sqrt(8)) xx (2sqrt(6))/(2sqrt(27) - sqrt(8)) =>
(2sqrt(6)(2sqrt(27) + sqrt(8)))/((2sqrt(27))^2 - (sqrt(8))^2) =>
((2sqrt(6) * 2sqrt(27)) + (2sqrt(6) * sqrt(8)))/((4 * 27) - 8) =>
(4sqrt(6)sqrt(27) + 2sqrt(6)sqrt(8))/(108 - 8) =>
(4sqrt(6 * 27) + 2sqrt(6 * 8))/100 =>
(4sqrt(162) + 2sqrt(48))/100 =>
(4sqrt(81 * 2) + 2sqrt(16 * 3))/100 =>
(4sqrt(81)sqrt(2) + 2sqrt(16)sqrt(3))/100 =>
((4 * 9sqrt(2)) + (2 * 4sqrt(3)))/100 =>
((color(red)(cancel(color(black)(4))) * 9sqrt(2)) + (2 * color(red)(cancel(color(black)(4)))sqrt(3)))/(color(red)(cancel(color(black)(100)))25) =>
(9sqrt(2) + 2sqrt(3))/25
