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Question

Prove that root of p+root of q is irrational


Solution

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 

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