Question

Show that `root(3)(12)` is an irrational number.

Answer 1

Given: Prove that `root(3)(12)` is an irrational number.

Step-wise calculation:

1. Assume, for the sake of contradiction, that `root(3)(12)` is rational.

Then it can be expressed as `root(3)(12) = a/b`, where (a) and (b) are integers, b ≠ 0 and gcd(a, b) = 1.

2. Cube both sides:

`12 = a^3/b^3`

⇒ a³ = 12b³

3. Since 12 = 2² × 3, the prime factors on the right side are 2 and 3. 

Thus, a³ is divisible by 2 and 3, which means (a) is divisible by both 2 and 3 because if a prime divides a cube, it divides the base.

So, (a) is divisible by 6.

4. Let a = 6c.

Substitute back:

(6c)³ = 12b³

⇒ 216c³ = 12b³

⇒ 18c³ = b³

5. Thus, b³ = 18c³ = 2 × 3² × c³.

6. From the above, b³ is divisible by 2 and 3, so (b) is divisible by 6.

7. Since both (a) and (b) are divisible by 6, it contradicts the initial assumption that gcd(a, b) = 1.

Our assumption that `root(3)(12)` is rational leads to a contradiction.

Hence, `root(3)(12)` is irrational.