Question

if   `x= 3 + sqrt8`, find the value of `x^2 + 1/x^2`

Answer 1

We know that `x^2 + 1/x^2 = (x +1/x)^2 - 2`. We have to find the value of `x^2 + 1/x^2`. As `x = 3 + sqrt8`

therefore

`1/x = 1/(3 + sqrt8)`

We know that rationalization factor for `3 + sqrt8` is `3 - sqrt8`. We will multiply numerator and denominator of the given expression `1/(3 = sqrt8)` by `3 - sqrt3` to get

`1/x = 1/(3 +  sqrt8) xx (3 - sqrt8)/(3 -sqrt8)`

`= (3 - sqrt8)/(9 - 8)`

`= 3 - sqrt8`

Putting the vlaue  of x and 1/x, we get

`x^2 + 1/x^2 = (3 + sqrt8 + 3 - sqrt8)^2 - 2`

`= (6)^2 - 2`

= 36 - 2

= 34

Hence the given expression is simplified to 34.