Question

prove that root √p and√q is an irrational,where p and q are primes​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

Here goes the solution!

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 = x^2

p + 2√pq + q = x^2

2√pq = x^2 - p - q

√pq = ( x^2 - p - q )/2

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

So, ( x^2 - 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 and √q is irrational, where p and q are distinct primes.