Question

given that √2is irrational . prove that 5\+3√2 is an irrational

Answer 1

Let 5/+3√2 be rational number
Now let 5/+3√2 = a/b
= 3√2=a-5b/b
Here we know that a-5b/b is rational number therefore 5/+3√2 is also rational no. But we have √2 as irrational no. Therefore our assumption is wrong and 5/+3√2 is irrational no.
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Answer 2

let us assume on the contrary that 5+3√2 is rational. then there exist co prime positive integers a and b such that
5+3√2=a/b
2√5=a/b-3
√5=a-3b/2b
√5 is irrational