Question

Show that: `x^3 + 1/x^3 = 52`, if x = 2 + `sqrt3`

Answer 1

Given: x = 2 + `sqrt3`

`1/x = 1/(2 + sqrt3) xx (2 - sqrt3)/(2 - sqrt3)`

`= (2 - sqrt3)/((2)^2 - (sqrt3)^2)`

`= (2 - sqrt3)/(4 - 3)`

`= 2 - sqrt3`

Now,

`x + 1/x = 2 + cancel(sqrt3) + 2 - cancel(sqrt3)`

`x + 1/x = 2 + 2`

`x + 1/x`= 4

`therefore x^3 + 1/x^3`

`= (x + 1/x)^3 - 3 * cancel(x) * 1/cancel(x) (x + 1/x)`

`= (4)^3 - 3 xx 4`

= 64 - 12

= 52