**1.** Prove that is irrational.

**Solution:**

Lets prove irrational by contradiction.

Lets suppose that is rational. It means that we have co-prime integers

**a and b **() such that .

Squaring both sides, we get

**(1)**

It means that 5 is factor of

Hence, **5 is also factor of** **a** by Theorem. **(2)**

If, **5 is factor of a**, it means that we can write **a=5c** for some integer **c**.

Substituting value of **a **in **(1) **, we get

It means that 5 is factor of . Hence, **5 is also factor of b** by Theorem. **(3)**

From **(2)** and **(3)**, we can say that **5** is factor of both **a and b**. But, **a and b** are **co-prime**. Therefore, our assumption was wrong. cannot be rational. Hence, it is irrational.