Question

Prove 5+6√3 is an irrational number

Answer 1

We have to show that 5+6√3 is an irrational number.

First of all, we have to prove three things :

(i) √3 is an irrational number. (You should prove this statement at the beginning of your proof.)

(ii) The sum of a rational and an irrational number is irrational

(iii) The product of a non-zero rational number and an irrational number is an irrational number.

The proofs of (ii) and (iii) are shown in the pictures attached with this answer. So, now after proving these three statements, we will come to our original proof :

PROOF :-

√3 is an irrational number

So, 6√3 is also an irrational number (Because, The product of a non-zero rational number and an irrational number is an irrational number.)

Therefore, 5+6√3 is a irrational number (Because, The sum of a rational and an irrational number is irrational)

Hence, 5+6√3 is a irrational number [proved]