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Question
If x = `(7 + 4sqrt(3))`, find the values of
`x^3 + (1)/x^3`
Solution
`x^3 + (1)/x^3`
`(x^3 + (1)/x^3) = (x + (1)/x)^3 - 3(x + (1)/x)` ----(1)
we will first find out `x + (1)/x`
`x + (1)/x = (7 + 4sqrt(3)) + (1)/((7 + 4sqrt(3))`
= `((7 + 4sqrt(3))^2 + 1)/((7 + 4sqrt(3))`
= `(49 + 48 + 56sqrt(3) + 1)/((7 + 4sqrt(3))`
= `(98 + 56sqrt(3))/((7 + 4sqrt(3))`
= `(14(7 + 4sqrt(3)))/((7 + 4sqrt(3))`
= 14
substitutingin (1)
`(x^3 + (1)/x^3) = (x + (1)/x)^3 -3(x + (1)/x)`
= (14)3 - 3 x 14
= 2744 - 42
= 2702
∴ `(x^3 + (1)/x^3)` = 2702
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