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Prove that sqrtp+sqrtq is an irrational,...

Prove that `sqrtp+sqrtq` is an irrational, where `p and q` are primes.

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Let possible `sqrt(p)+sqrt(q)` is rational.
`:. sqrt(p)+sqrt(q) =a` for some rational number 'a'
`rArr " "sqrt(p) =a -sqrt(1) rArr (sqrt(p))^(2)=(a-sqrt(q))^(2)`
`rArr " "p=a^(2) +q-2asqrt(a)rArr 2asqrt(q)=a^(2) +q-p`
`rArr " "sqrt(q)=(a^(2) +q-p)/(2a)`
`:. q ` is prime.
`:. sqrt(q)` is irrational.
But R.H.S. i.e, `(a^(2)+q-p)/(2a)` is rational and so. L.H.S. cannot be equal to R.H.S.
Which contradicts our assumption.
Hence `sqrt(p) +sqrt(q)` is irrational .
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