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Hi there! Have questions about your homework? At Brainly, there are 60 million students who want to help each other learn. Questions are usually answered in less than 10 minutes. Try it for yourself by posting a question! :D

First, we'll assume that √p + √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 and 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 a contradiction. Original assumption must be wrong.

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

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We can also show that √p + √q is irrational, where p and q are non-distinct primes, i.e. p = q

We use same method: Assume √p + √q is rational. √p + √q = x, where x is rational √p + √p = x 2√p = x √p = x/2

Since both x and 2 are rational, and rational numbers are closed under division, then x/2 is rational. But since p is not a perfect square, then √p is not rational. But this is a contradiction. Original assumption must be wrong.

So √p + √q is irrational, where p and q are non-distinct primes