Question

If `x^4 + 1/x^4 = 2, "find the value of"  x^2 + 1/x^2, x + 1/xandx^3 + 1/x^3.`

Answer 1

Here, `x^4 + 1/x^4 = 2,`

Let’s find `x^2 + 1/x^2,`

Using the identity:

`(x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2`

`(x^2 + 1/x^2)^2 = 2 + 2`

`(x^2 + 1/x^2)^2 = 4`

`x^2 + 1/x^2 = sqrt4`

`∴`x^2 + 1/x^2 = 2`

Let’s find `x + 1/x,`

Using the identity:

`(x + 1/x)^2 = x^2 + 1/x^2 + 2`

`(x + 1/x)^2 = 2 + 2`

`(x + 1/x)^2 = 4`

`x + 1/x = +-sqrt4`

∴ `x + 1/x = +-2`

Let’s find `x^3 + 1/x^3,`

Using the identity:

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

`(2)^3 = x^3 + 1/x^3 + 3(2)`

`8 = x^3 + 1/x^3 + 6`

`x^3 + 1/x^3 = 8-6`

∴ `x^3 + 1/x^3 = +-2`

Hence, the values of `x^2 + 1/x^2, x + 1/xandx^3 + 1/x^3` are 2, ±2, and ±2.