Let 2+3√5 be rational.
Then, 2+3√5 = p/q
=> 3√5 = p/q - 2
=> √5 = p-2q/3q
Here, √5 is irrational (given) but p-2q/3q is rational. As rational ≠ irrational, 2+3√5 is irrational.
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Answers & Comments
Let 2+3√5 be rational.
Then, 2+3√5 = p/q
=> 3√5 = p/q - 2
=> √5 = p-2q/3q
Here, √5 is irrational (given) but p-2q/3q is rational. As rational ≠ irrational, 2+3√5 is irrational.