Prove or disprove the cube root of 600 is an irrational numb

Solution

Answer :

Assume that (600) is not irrational.That is assume that  (600) is rational number.

So , we can write  (600) as  (600) = a/b for some a,b Z⁺ and gcd(a,b) = 1.

Take cube on both sides , we get

600 = a³/ b³

a³ = 600b³   implies that 600 | a³

      implies that 600 | a

implies that a = 600c for some a,c Z

If you then plug this back into the equation, you would find that 600 divides b as well, which is a contradicting to our assumption that a and b are coprime.

Hence ,  (600) is irrational