Question

Show that `root(3)(4)` is an irrational number.

Answer 1

Given: Show that `root(3)(4)` the cube root of 4 is irrational.

Step-wise calculation:

1. Assume `root(3)(4)` is rational.

2. Thus, it can be written as `root(3)(4) = a/b`, where (a) and (b) are integers with no common factors other than 1 and b ≠ 0.

3. Cube both sides:

`4 = a^3/b^3`

⇒ a³ = 4b³

4. From this equation, we see that a³ is divisible by 2 since 4 = 2² and so 4b³ includes factors of 2.

5. Let a = 2c, where (c) is an integer.

6. Substitute back:

(2c)³ = 4b³ 

⇒ 8c³ = 4b³

⇒ 2c³ = b³

7. Hence, b³ = 2c³, so b³ is divisible by 2, meaning (b) is divisible by 2.

8. Thus, both (a) and (b) are divisible by 2, which contradicts the assumption that (a) and (b) have no common factors other than 1.

9. Therefore, the assumption that `root(3)(4)` is rational is false.