Question

How do you rationalize the denominator and simplify `1/(1+root3x+root3(x^2)`#?

Answer 1

Refer to explanation

Explanation:

Set `t=root(3) x` hence we have that

`1/(1+t+t^2)`

We know that `t^3-1=(t-1)*(1+t+t^2)=>1+t+t^2=(t^3-1)/(t-1)`

hence

`1/(1+t+t^2)=1/((t^3-1)/(t-1))=(t-1)/(t^3-1)`

because `t=root(3) x`

we replace it and get

`(root(3) x -1)/(((root(3) x)^3)-1)=(root(3) x-1)/(x-1)`