Question

if a,b primes then square root a+square root b​

Answer 1

Step-by-step explanation:

If p is a prime number, then p–√ is irrational.

I know that this question has been asked but I just want to make sure that my method is clear. My method is as follows:

Let us assume that the square root of the prime number p is rational. Hence we can write p–√=ab. (In their lowest form.) Then p=a2b2, and so pb2=a2.

Hence p divides a2, so p divides a. Substitute a by pk. Find out that p divides b. Hence this is a contradiction as they should be relatively prime, i.e., gcd(a,b)=1.