if an is irrational then a is irrational


If a be a real number and n is an integer such that an is irrational, then a is irrational.


We show this by way of contrapositive. In other words, we show that, if a is rational, then an is rational.

Let a be rational. Then there exist integers b and c with c0 such that a=bc. Thus, an=bncn, which is a rational numberPlanetmathPlanetmath. ∎

Note that the converseMathworldPlanetmath is not true. For example, 2 is irrational and (2)2=2 is rational.

Title if an is irrational then a is irrational
Canonical name IfAnIsIrrationalThenaIsIrrational
Date of creation 2013-03-22 14:18:50
Last modified on 2013-03-22 14:18:50
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11J82
Classification msc 11J72