Question

Prove that 2+5√3 is an irrational number, given that √3 is an irrational number

Answer 1

here is your answer

since, 2+5√3 is rational number

so, p/q = 2+5√3 (where p and q are two integer having no common factor other then 1 and q equal not 0)

so, p/q = 2+5√3

then p/q -5 = 2√3

p -5q/q = 2√3

p -5q/ 2q =√3-------------------------- ( 1 )

rational = irrational ( which is impossible )

so from first

2 +5√3 is an irrational number

hence proved

Answer 2

Latest assume that 2 + 5√3 is an rational number,

Thus 2 + 5√3 can be written in the form p/q where,

(p,q are coprime numbers

p,q ≈ integers and q ≠0)

2 + 5√3 = p/q

5√3 = p/q -2

5√3 = (p-2q)/q

√3 = (p-2q)/5q

Since RHS is rational then LHS should also be rational,

But this contradicts (opposes) the fact that root 3 is irrational.

This contradiction has arisen due to our incorrect assumption that 2+5√3 is rational.

Hence proved that 2 + 5√ 3 is an irrational number.