Question

Prove that 7+3√2 is irrational​

Answer 1

Answer:

This contradiction arise due to our wrong assumption. Hence, 7+3√2 cannot be a rational number, ∵ it is an irrational number.

Answer 2

Answer:

let us assume that 7+3√2 is rational

7+3√2=p/q(p and q are coprimes)

3√2=p/q-7

√2=p-7q/3q

for this we need to see if √2 is rational or not

let's assume that √2 is rational

√2 = p/q

Here p and q are coprime numbers and q ≠ 0

Solving

√2 = p/q

On squaring both the side we get,

=>2 = (p/q)2

=> 2q2 = p2……………………………..(1)

p2/2 = q2

So 2 divides p and p is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ………………………………..(2)

From equations (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, p,q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√2 is an irrational number.

hope this helps