Question

prove that 2+5√3 is irrational​

Answer 1

Step-by-step explanation:

Let 2 + 5√3 = a, where a is a rational number. Which is a contradiction as LHS is irrational and RHS is rational. ∴ 2 + 5√3 can not be rational. Hence, 2 + 5√3 + is irrational.

Answer 2

Step-by-step explanation:

Let 2 + 5√3 = a, where a is a rational number.

√3 = (a – 2)/5

Which is a contradiction as LHS is irrational and RHS is rational.

∴ 2 + 5√3 can not be rational.

Hence, 2 + 5√3 + is irrational.

Alternate Method

Let 2 + 5√3 be rational.

∴ 2 + 5√3 = p/q ; p, q are integers, q ≠ 0

=> √3 = (p/q - 2) ÷ 5

√3 = (p - 2q)/5q

LHS is irrational and RHS is rational which is a contradiction.

∴ 2 + 5√3 is irrational.

Hence proved