Question

If a is prime number then prove that underoot a is irrational

Answer 1

Let us assume, to the contrary, that √a is 
rational. 
So, we can find coprime integers p and q(q ≠ 0) 
such that √a = p/q
=> √a q = p 
=> aq2 = p2 ….(i) [Squaring both the sides] 
=> p2 is divisible by a 
=> p is divisible by a 
So, we can write p = ac for some integer c. 
Therefore, p2 = a2c2 ….[Squaring both the sides] 
=> ab2 = a2c2 ….[From (i)] 
=> b2 = ac2 
=> b2 is divisible by a 
=> b is divisible by a
=> a divides both p and q. 
=> p and q have at least a as a common factor. 
But this contradicts the fact that p and q are coprime. 
This contradiction arises because we have 
assumed that √a is rational. 
Therefore, √a is irrational.