Question

Prove that 8-√7 is an irrational number​

Answer 1

Answer with Step-by-step explanation:

Let us assume that 8 -√7 is a rational number.

So it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

8 - √7 = p/q

Here p and q are coprime numbers and q ≠ 8 - √7 = p/q

On squaring both the side we get,

⇒ 57 = (p/q)²

⇒ 57 q² = p²              ..(1)

p²/57 = q²

So, 57 divides p and q but, p and q are multiple of 57.

⇒ p = 57m

⇒ p² = 3249m²          ..(2)

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

57q² = 3249m²

⇒ q² = 57m²

⇒ q² is a multiple of 57

⇒ q is a multiple of 57

∴  p, q have a common factor 57. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

8 - √7 is an irrational number.