Question

if m and n are prime positive integer prove that root M + root N is a irrational number​

Answer 1

$<b>$Given:-

M and N are positive primes

To prove:-

√M + √N is an irrational number.

Proof :-

Let √M + √N be a Rational number

Therefore, √M + √N = p/q

==> ( √M + N ) ² = p ²/q ²

[ Squaring on both sides ]

==> M + N + 2 √MN = p ²/q ²

==> 2 √MN = p ²/q ² - M - N

==> 2 √MN = p ² - Mq ² - Nq ²/q ²

==> √MN = p ² - Mq ² - Nq ²/2q ²

Now, √MN which is an irrational number as M and N are primes is equal to a Rational number where ( p ≠ 0, q ≠ 0 , M ≠ 0, N ≠ 0 ) is a contradiction.

This contradiction has arisen because of our assumption that √M + √N is a rational number, which is incorrect .

Hence, √M + √N is an irrational number [ proof by contradiction ].

$\Huge\yellow{Thank You}$

Answer 2

Answer:

Answer. √M + √N is an irrational number.

If n is a perfect square then √n is a an integer and therefore rational, so it suffices to prove that if n is not a perfect square, then √n is irrational