Question

prove that root2+3/root2 is irrational

Answer 1

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.

Answer 2

Proof : Let (√2 + 3)/√2 be a rational number which is equal to x.

⇒ x = (√2 + 3)/√2

⇒ x = (√2 + 3)/√2 × √2/√2

⇒ x = (2 + 3√2)/2

⇒ 2x = 2 + 3√2

⇒ (2x - 2)/3 = √2

⇒ 2(x - 1)/3 = √2

Thus √2 is in p/q form and hence a rational number. This leads to contradiction for a fact that √2 is actually an irational number.

That's why (√2 + 3)/√2 is an irractional.

Q.E.D