Question

Prove that (√3-7)² is irrational

Answer 1

(√3-7)²

→ √3²+7²-2(√3)(7)

→ 3+49-14√3

→ 51-14√3

Let 51-14√3 be a rational number.

A rational number can be written in the form of p/q.

51-14√3 = p/q

14√3 = 51-p/q

14√3 = (51q-p)/q

√3 = (51q-p)/14q

p,q are integers then (51q-p)/14q is a rational number.

Then √3 is also a rational number.

But this contradicts the fact that √3 is an irrational number.

Our supposition is false.

Therefore, (√3-7)² is an irrational number,

Hence proved.