Question

if n is positive integer which is not a perfect square. check whether √n is rational or irrational?​

Answer 1

Answer:

Since n is not a perfect square, there exists at least one prime number p such that

n=pαq

where q∈N is coprime to p and α≥1 is odd.

Now, suppose that n is a rational number, namely,

root n=ba

where a,b are natural numbers and they are coprime to each other.

Then, we have

n=b2a2⇒a2n=b2.

By the fundamental theorem of arithmetic, this implies that the number of p in the left hand side is odd, and that the number of p in the right hand side is even. This is a contradiction.

Hence, root n is an irrational number

Answer 2

Answer:

Step-by-step explanation: