Question

If q is a prime numbers, then prove that root q is an irrational number.

Answer 1

Answer:

Step-by-step explanation:

If possible,let √q be a rational number.

also a and b is rational.

then,√q = a/b

on squaring both sides,we get,

(√q)²= a²/b²

→q = a²/b²

→b² = a²/q [p divides a² so,q divides a]

Let a= qr for some integer r

→b² = (qr)²/q

→b² = q²r²/q

→b² = qr²

→r² = b²/q [p divides b² so, q divides b]

Thus q is a common factor of a and b.

But this is a contradiction, since a and b have no common factor.

This contradiction arises by assuming √q a rational number.

Hence,√q is irrational.

Answer 2

Answer:

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