Question

Show that `root(3)(5)` is an irrational number.

Answer 1

Given: Show that `root(3)(5)` cube root of 5 is an irrational number.

Stepwise calculation:

1. Assume, for the sake of contradiction, that `root(3)(5)` is rational.

That means we can write `root(3)(5) = a/b`, where (a) and (b) are integers with no common factors co-prime and b ≠ 0.

2. Cube both sides: `5 = (a/b)^3 = a^3/b^3`

Multiplying both sides: 5b³ = a³

3. From 5b³ = a³, it follows that a³ is divisible by 5.

Therefore, (a) must be divisible by 5.

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

4. Substitute a = 5c back into the equation: 

5b³ = (5c)³ = 125c³ 

Simplify: 5b³ = 125c³

⇒ b³ = 25c³

5. From b³ = 25c³, b³ is also divisible by 5, so (b) is divisible by 5.

6. Therefore, both (a) and (b) are divisible by 5, contradicting the assumption that `a/b` was in simplest terms co-prime.

Since the assumption that `root(3)(5)` is rational leads to a contradiction, `root(3)(5)` is irrational.

Thus, the cube root of 5 is an irrational number.