We have to prove that 1/3 + 2√5 irrational.

We know 2√5 is irrational   ...........1

Let 1/3 + 2√5 be reational.

So, we can write it in the form of p/q where p and q are co prime and q is not equal to zero

1/3 + 2√5 = p/q

2√5 = p/q - 1/3

2√5 =  (3p - q)/ 3q

√5 = (3p - q)/ 6q   .......2

Here, √5 should be rational from the equation 2.

But we know that √5 is irrational.

So, our assumption is incorrect.

Hence, 1/3 + 2√5 is irrational.