Question

If x = `((sqrt(3) + 1))/((sqrt(3) - 1)` and y = `((sqrt(3) - 1))/((sqrt(3) + 1)`, find the values of 

x³ + y³

Answer 1

x³ + y³

x³ + y³ = (x + y)³ - 3xy(x + y)  ----(1)

∴ (x + y) = `((sqrt(3) + 1))/((sqrt(3) - 1)) + ((sqrt(3) - 1))/((sqrt(3) + 1)`

= `((sqrt(3) + 1)^2 + (sqrt(3) - 1)^2)/(3 - 1)`

= `(3 + 1 + 2sqrt(3) + 3 + 1 - 2sqrt(3))/(2)`

= `(8)/(2)`

= 4

and xy = `((sqrt(3) + 1))/((sqrt(3) - 1)) xx ((sqrt(3) - 1))/((sqrt(3) + 1)`

= `(3 - 1)/(3 - 1)`

= 1

substitutingin (1), we get
x³ + y³
= (x + y)³ - 3xy(x + y)
= 64 - 3 x 4
= 64 - 12
= 52