Solution:
∵p,q,r,∈Q and √p+√q+√r∈Q
⇒(√p+√q+√r)2∈Q ⇒√pq+√qr+√rp∈Q
Case I: Let exactly one of √p,√q,√r is irrational
WLOG, √p,∉Q but √r,√q∈Q
From (i), √p(√q+√r)∈Q (contradiction)
Case II: Let exactly two out of √p,√q,√r are irrational.
WLOG, √p,√q∉Q but √r∈Q
From(i), √p√q+√r√p+√r√q+(√r)2∈Q
⇒(√p+√r)(√q+√r)∈Q
∵ both √p+√r and √q+√r are irrational, hence they must be conjugate of each other. (Contradiction)
Case III: Let all √p,√q,√r are irrational.
Let √p+√q+√r=x when x∈Q+
From (i), √p(x−√p)+√q√r∈Q
⇒x√p+√q√r∈Q (Contradiction)
Hence, all √p,√q and √r must be rational.
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