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