Question

let a,b,c and d be positive rational such that a+root b =c +root d then prove that either a=c b=d or b and d are square of rational.
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Answer 1

1) a,b,c and d are positive rational numbers .

a + √b = c + √d
a - c = √d - √b

Here ,
LHS ( a-c) is rational and therefore RHS has to be rational .
√b-√d has to be rational
implies either √b = √d and a = c
or b and d are perfect square of a rational number.