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Question
If `x^2 + 1/x^2` = 23, then find the value of `x + 1/x` and `x^3 + 1/x^3`
Solution
`x^2 + 1/x^2` = 23
`(x + 1/x)^2 - 2` = 23 ...[a2 + b2 = (a + b)2 − 2ab]
`(x + 1/x)^2` = 23 + 2
⇒ `(x + 1/x)^2` = 25
`x + 1/x = sqrt(25)`
`x+ 1/x` = ± 5
`x^3 + 1/x^3 = (x + 1/x)^3 - 3x xx 1/x(x + 1/x)`
When x = 5 ...[a3 + b3 = (a + b)3 – 3ab(a + b)]
= (5)3 – 3(5)
= 125 – 15
= 110
when x = – 5
`x^3 + 1/x^3` = (–5)3 – 3(–5)
= – 125 + 15
= – 110
∴ `x^3 + 1/x^3` = ± 110
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