5. Prove that v7 is an irrational number.

1.	5. Prove that v7 is an irrational number.
Answer» Let √7 be rationalLet √7 = a / b wher a,b are integers b ≠ 0we also suppose that a / b is written in the simplest form Now √7 = a / b ⇒ 7 = a²/b² ⇒ 7b²= a² ∴ 7b²is divisible by 7 ⇒ a²is divisible by 7 ⇒ a is divisible by 7 ∴ let a = 7c a² = 49c² ⇒ 7b²= 49c² ⇒b²= 7c² ∴ 7c² is divisible by 7 ∴ b² is divisible by 7 ∴ b is divisible by 7 ∴a are b are divisible by 7this contradicts our supposition that a/b is written in the simplest formHence our supposition is wrong ∴ √7 is irrational number.

Answer»

Let √7 be rationalLet √7 = a / b wher a,b are integers b ≠ 0we also suppose that a / b is written in the simplest form

Now √7 = a / b

⇒ 7 = a²/b² ⇒ 7b²= a²

∴ 7b²is divisible by 7

⇒ a²is divisible by 7

⇒ a is divisible by 7

∴ let a = 7c

a² = 49c²

⇒ 7b²= 49c²

⇒b²= 7c²

∴ 7c² is divisible by 7

∴ b² is divisible by 7

∴ b is divisible by 7

∴a are b are divisible by 7this contradicts our supposition that a/b is written in the simplest formHence our supposition is wrong

∴ √7 is irrational number.

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