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Question
Given that `sqrt2` is irrational prove that `(5 + 3sqrt2)` is an irrational number
Solution
Let us assume, to the contrary that `(5 + 3sqrt2)` is rational.
That is, we can find coprime a and b `(b != 0)` such that `5 + 3sqrt2 = a/b`
Therefore `5 - a/b = 3 sqrt2`
`=> 5/3 - a/(3b) = sqrt2`
`=> (5b -a)/(3b) = sqrt2`
Since, a and b are integers, we get `(5b - a)/(3b)` is rational and so `sqrt2` is rational
which contradicts the fact that `sqrt2` is irrational
So our assumption was wrong that `(5 + 3sqrt2)` is rational
Hence we conclude that `5 + 3sqrt3` is irrational
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