- #1

- 308

- 0

## Homework Statement

Prove that [tex]e^{\frac{n}{m}}[/tex] is trancendental, where m>0 and n are integers.

## Homework Equations

e is trancendental

## The Attempt at a Solution

A hint I got said this:

"Let ln(m\n)=m\n"

so I did this:

then[tex] \frac{m}{n} = e^{\frac{m}{n}}[/tex]

so then [tex](\frac{m}{n})^n=(e^{\frac{m}{n}})^n[/tex]

and then [tex](\frac{m}{n})^n-e^m=0[/tex]

this contradicts the trancendence of e, therefore e^n\m is trancendental.

I'm confused about the hint, and I'm not convinced that this is actually valid. This hint came from one of my classmates and aside from pulling out if the thin air, I don't see how he arrived at this.

Any input will be appreciated.

CC

Last edited: