Question

Prove that root2+3/root2 is an irrational no

Answer 1

root2+3/root 2 =root 2+3/root 2×root 2/root 2=2+2root 3/2 which is irrational

Answer 2

Answer:

To prove that √2 + 3/√2 is irrational.

Let us assume that √2 + 3/√2 is rational.

√2 + 3/√2 = a/b, where 'a' and 'b' are integers and 'b' ≠ 0.

Squaring both sides,

(√2 + 3)^2/(√2)^2 = (a/b)^2

(√2 + 3)^2/2 = a^2/b^2

2 + 9 + 6√2/2 = a^2/b^2

11 + 6√2 = 2a^2/b^2

6√2 = 2a^2/b^2 - 11

LCM for RHS: b^2

6√2 = 2a^2 - 11b^2/b^2

√2 = 2a^2 - 11b^2/6b^2

Since 'a' and 'b' are integers, 2a^2 - 11b^2/6b^2 is rational.

⇒ √2 is rational.

This contradicts the fact that √2 is irrational.

This contradiction has arisen due to our wrong assumption.

Therefore, √2 + 3/√2 is irrational.