Question

prove that root P + q where p u l Prime​

Answer 1

Answer:

....

Step-by-step explanation:

first we'll assume that √p and √q is rational , where p and q are distinct primes √p+√q=x, where x is rational

Rational numbers are closed under multiplication, so if we square both sides, we still get rational numbers on both sides.

(√p+√q)²=x²

p+2√pq+q=x²

2√pq=x²-p-q

√pq= (x²-p-q)/2

Now, x, x²,p,q & 2 are all rational, and rational numbers are closed under subtraction and division.

so,(x²-p-q)/2 is rational

But since p and q are both primes, then pq is not a perfect square and therefore

√pq is not rational. But this is contradiction. Original assumption must be wrong.

So, √p & √q is irrational, where p and q are distinct primes.