Show that is not a rational number.

Let be a rational number, say where q ≠ 0.
Then   =
Since 1³ = 1 , and 2³ = 8, it follows that 1 <  < 2
Then q > 1 because if q = 1 then will be an integer, and there is no integer between 1 and 2.
Now, 6 = 
  6 = 
6q² = 
q being an integer, 6q² is an integer, and since q > 1 and q does not have a common factor with p and consequently with p³. So, is a fraction, different from an integer. 
Thus 6q² ≠ .
This contradiction proves the result.