Question

Show that `root(3)(2)` is an irrational number.

Answer 1

Given: We want to show that `root(3)(2)` the cube root of 2 is an irrational number.

Step wise calculation:

1. Assume `root(3)(2)` is rational.

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

2. Cube both sides:

`2 = a^3/b^3`

⇒ a³ = 2b³

3. Since a³ = 2b³, a³ is even because it is twice another integer.

4. If a³ is even, (a) must be even because the cube of an odd number is odd.

5. Let a = 2c for some integer (c).

6. Substitute back into the equation:

(2c)³ = 2b³

⇒ 8c³ = 2b³

⇒ 4c³ = b³

7. This implies b³ is also even, so (b) is even.

8. Hence, both (a) and (b) are even, contradicting the initial assumption that `a/b` is in simplest form with no common factors other than 1.

Our initial assumption that `root(3)(2)` is rational leads to a contradiction.

Therefore, `root(3)(2)` is irrational.