Question

prove that root2 + 3/root2 is irrational

Answer 1

let us assume ,to the contrary ,that 2+3√5 is rational. that is ,we find comprime a and b (b not equal to 0) such that
2+3√5 =a/b
therefore. 3√5=a/b-2
or. √5=a-2b
..... . ......... 3b

since 2,3,a and b are integers, therefore √5 is rational and a-2b is rational. 3b
but this contradict the fact that √5 is irrational so, we conclude the fact that 3√5 is irrational. hence ,2+3√5 is irrational.

hope this help ...

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.