Question

Prove that
√
3
+
2
√
3
is irrational

Answer 1

Hey mate here both are irrational number and sum of irrational is always irrational hence it is irrational number.

Proved

Answer 2

Let us suppose that √3 is rational. Then there exist two positive integers a and B such that

√3 = a/b

Where a and B are co primes

Squaring on both side gives us

3=a^2/b^2

35b^2 = a^2

It means 3 is a factor of a^2 and a as well

3c = a. (as 3 is a factor of a)

Squaring on both sides gives us

9c^2 = a^2

9c^2 = 3b^2. ( As proved above)

b^2 = 3c^2

It means 3 is also a factor of B.

Hence it is a contradiction as a and b were co primes.

Hence our supposition is wrong and √3 is irrational.

For further solution see the pic