Application of Fermat's last theorem

So here's my issue.

I'm not trying to write a formal proof, I'm just trying to find all cases for this problem.

Question: Assuming Fermat's Last Theorem (FLT), show that x^n+y^n=1 (n integer >2), contains no points with rational coordinates except those points where the curve crosses the axis.

I have graphed the equation for an odd n and an even n. They both have a distinct shape so I think I need two main cases where n is even or odd with subcases for x,y being <0 and >0. It's easy to show that there are rational coordinates at the intersection of the axes.

I then solved the equation for y. I thought it would be possible to show that if x was rational (positive or negative) that y=(1-x^n)^(1/n) makes y irrational. While I've read most nth roots are irrational, this doesn't seem conclusive enough to say y will always be irrational.

Any help or direction to conquering this problem?

Thanks in advance

Edit:

It can also be noted that for

n->odd : -∞ <x < ∞ -∞ < y < ∞

n->even -1< x < 1 -1 < y < 1