Question

Prove that root 5 is a irrational number ​

Answer 1

Answer:

Refer the attachment. Hope this helps you

Answer 2

if √5 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean:

(a/b)² = 5. Squaring,

a² / b² = 5. Multiplying by b²,

a² = 5b².

If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors. 5b² has one more prime factor than b², meaning it would have an odd number of prime factors.

Every composite has a unique prime factorization and can't have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so √5 cannot be rational. It is, therefore, irrational.