Question

Prove the following are irrational numbers
$3+\sqrt{5}$

Answer 1

Let us assume $3+(5{)}^{1/2}$ is a rational number

we know that 3 is a rational number.

Now subtract 3 from $3+(5{)}^{1/2}$ and we know that the difference of two rational number is also a rational number

That means, $3+(5{)}^{1/2}-3=(5{)}^{1/2}$ should be a rational number but it is not

This means that

$3+(5{)}^{1/2}$ is an irrational number.

Hence Proved!!