Question

Prove that √p + √qis irrational, where p and q are primes.​

Answer 1

Step-by-step explanation:

First, we'll assume that

pand q is rational , where p and q are distinctprimes

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=x p+2

pq+q=x 2 pq =x −p−q

pq = 2(x 2−p−q)

Now,

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

So,

2(x2−p−q) is rational.

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

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

So,

p and q is irrational, where p and q are distinct primes.

Answer 2

Answer:

its easy and short method than above one

Step-by-step explanation:

let √p+√q be a rational number

we known that every rational number is in the form of a/b

√P +√Q= a/b

√p=a/b-√q

square on both side

p=a²/b²-2a/b√q+p

and √q=a²b+b³(q-p)/2a

hence it is rational number

But q be prime so the √q is irrational number

hence (√p+√q) is a irrational number

please mark me as brainlist