Question 762576
This a lot of multiplying, but here goes. Start with 
(1) {{{x = (3 + sqrt(5))/(3 - sqrt(5))}}}
Now multiply numerator and denominator of (1) by
(2) {{{(3 + sqrt(5))}}} and get
(3) {{{x = (9 + 6sqrt(5) +5)/4}}} or
(4) {{{x = (14 + 6sqrt(5))/4}}} or
(5) {{{x = (7 + 3sqrt(5))/2}}}
Now get
(6) {{{x^2 = (49+42sqrt(5)+45)/4}}} or
(7) {{{x^2 = (94+42sqrt(5))/4}}}
Now get {{{x^3}}} from
(8) {{{x^3 = x*x^2}}} or
(9) {{{x^3 = (7 + 3sqrt(5))*(94+42sqrt(5))/8}}} or
(10) {{{x^3 = (7*94 + (3*94 + 7*42)*(sqrt(5))+ 3*42*5)/8}}} or
(11) {{{x^3 = (658 + 576(sqrt(5))+ 630)/8}}} or
(12) {{{x^3 = 161 + 72sqrt(5)}}} 
From the original problem we see that
(13) y = 1/x or
(14) {{{y^3 = 1/x^3}}} or
(15) {{{y^3 = 1/(161+72sqrt(5))}}}
Now add (12) and (15) to get the desired sum
(16) {{{x^3+y^3= 161 + 72sqrt(5) +  1/(161+72sqrt(5))}}} or
(17) {{{x^3+y^3= (161^2 + 2*161*72sqrt(5)+ 72*72*5+1)/(161+72sqrt(5))}}} or
(18) {{{x^3+y^3= (51842 + 23184sqrt(5))/(161+72sqrt(5))}}}
The GCF of the numerator of (18) is 2*7*23 or 322. So we can write (18) as
(19) {{{x^3+y^3= 322*(161 + 72sqrt(5))/(161+72sqrt(5))}}} which reduces to
the final answer,
(20) {{{x^3+y^3= 322}}}