Question

State whether the following statement is true or not:

$(3+\sqrt{5})$ is an irrational number.

• A. True
• B. False

Answer 1

The correct option is A True

Let us suppose $3+\sqrt{5}$ is rational.

$=>3+\sqrt{5}$ is in the form of $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$

$=>\sqrt{5}=\frac{p}{q}-3$

​$=>\sqrt{5}=\frac{p-3q}{q}$

as $p,q$ and $3$ are integers $\frac{p-3q}{q}$ is a rational number.

$=>\sqrt{5}$ is a rational number.

But we know that $\sqrt{5}$ is an irrational number.

So this is a contradiction.

This contradiction has arisen because of our wrong assumption that $3+\sqrt{5}$ is a rational number.

Hence $3+\sqrt{5}$ is an irrational number.