Question

Prove that_/9 is an irrational number.​

Answer 1

Answer:

Let cube root 9 is an irrational number then it is of the form

cube root 9 = p/q where p and q are co prime numbers and q is not 0.

Then p3/q3 = 9

or p3 = 9q3.....(i)

or 9 is divisible by p3

or 9 is divisible by p.

Now let p = 9a

Cubing both sides we get

p3 = 729 a3.....(ii)

From (i) and (ii) we get

9q3 = 729a3

or q3 = 81a3

It implies 81 is divisible by q3

or 9 is divisible by q.

But this contradicts to the fact that p and q are co primes

Hence our supposition is wrong

Therefore cube root 9 is irrational.