section_path: Proof of irrationality for non-perfect n th power x
section_name: Proof of irrationality for non-perfect n th power x
paragraph_text: Proof of irrationality for non-perfect n th power x
  
  Assume that x n {\displaystyle {\sqrt{x}}} is rational. That is, it can be reduced to a fraction a b {\displaystyle {\frac {a}{b}}}, where a and b are integers without a common factor.
  
  This means that x = a n b n {\displaystyle x={\frac {a^{n}}{b^{n}}}}.
  
  Since x is an integer, a n {\displaystyle a^{n}} and b n {\displaystyle b^{n}} must share a common factor if b ≠ 1 {\displaystyle b\neq 1}. This means that if b ≠ 1 {\displaystyle b\neq 1}, a n b n {\displaystyle {\frac {a^{n}}{b^{n}}}} is not in simplest form. Thus b should equal 1.
  
  Since 1 n = 1 {\displaystyle 1^{n}=1} and n 1 = n {\displaystyle {\frac {n}{1}}=n}, a n b n = a n {\displaystyle {\frac {a^{n}}{b^{n}}}=a^{n}}.
  
  This means that x = a n {\displaystyle x=a^{n}} and thus, x n = a {\displaystyle {\sqrt{x}}=a}. This implies that x n {\displaystyle {\sqrt{x}}} is an integer. Since x is not a perfect n th power, this is impossible. Thus x n {\displaystyle {\sqrt{x}}} is irrational.