Is Every Root of 2 Higher Than 1 Irrational?

[Char. Limit]

I was thinking on the square root of 2 being irrational proof... and I got the idea that you could use the same idea for every root higher than two. The cube root, the quartic root, the quintic root, etcetera. (Obviously assuming the roots are natural numbers.)

As a reassurance I'm not crazy, is this correct?

 

[Hurkyl]

You are not crazy. At least for this particular reason.

 

[Char. Limit]

Good to know.

Is there a way to prove that other roots that aren't natural numbers are irrational?

 

[Hurkyl]

Prime factorization.

 

[Hurkyl]

Also, rational root^* theorem.

*: as in, root of a polynomial[/size]

 

[Char. Limit]

So... are all roots of two that aren't 1 irrational?

 

[HallsofIvy]

Almost. Of course the "1" root of 2, 2^1/1= 2 is rational! If n is a positive integer, greater than 1, then 2^1/n is irrational.