Is the cube root of 2 irrational?

Answer 1

The cube root of 2 is irrational. The proof that the square root of 2 is irrational may be used, with only slight modification.

Assume the cube root of 2 is rational. Then, it may be written as a/b, where a and b are integers with no common factors. (This is possible for all nonzero rational numbers). Since a/b is the cube root of 2, its cube must equal 2. That is,

(a/b)3 = 2

a3/b3 = 2

a3 = 2b3.

The right side is even, so the left side must be even also, that is, a3 is even. Since a3 is even, a is also even (because the cube of an odd number is always odd). Since a is even, there exists an integer c such that a = 2c. Now,

(2c)3 = 2b3

8c3 = 2b3

4c3 = b3.

The left side is now even, so the right side must be even too. The product of two odd numbers is always odd, so b3 cannot be odd; it must be even. Therefore b is even as well.

Since a and b are both even, the fraction a/b is not in lowest terms, thus contradicting our initial assumption. Since the initial assumption cannot have been true, it must be false, and the cube root of 2 is irrational.