Question

Prove that 2 + 5√3 is an irrational number.

Answer 1

Step-by-step explanation:

To prove:3 + 2√5 is an irrational number. Proof: Letus assume that 3 + 2√5 is a rational number.

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Prove that 3+2√5 is irrational.

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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.