Question

If `x + (1)/x = 5`, find the value of `x^2 + (1)/x^2, x^3 + (1)/x^3` and `x^4 + (1)/x^4`.

Answer 1

`x + (1)/x = 5`                 ...(1)

Squaring both sides of (1)

`(x + 1/x)^2` = (5)²

⇒ `x^2 + (1)/x^2 + 2` = 25

⇒ `x^2 + (1)/x^2`
= 25 - 2
= 23                             ...(2)

Cubing both sides of (1),

`(x  + 1/x)^3` = 95³

`x^3 + (1)/x^3 + 3 (x + 1/x)` = 125

⇒ `x^3 + (1)/x^3 + 3 (5)` = 125

⇒ `x^3 + (1)/x^3`
= 125 - 15
= 110

Squaring both sides of (2),

`(x^2 + 1/x^2)^2` = (23)²

⇒ `x^4 = (1)/x^4 + = 529`

⇒ `x^4 + (1)/x^4`
= 529 - 2
= 527.