Prove that the cube root4 is irrationalSolutionWe can prove

Prove that the cube root(-4) is irrational.

Solution

We can prove it by contradiction.

Suppose 4^(1/3) = a/b, where a and b are positive co-prime integers.

Then 4 = a^3 / b^3 -----> 2*2*b^3 = a^3.

So a^3 is even and hence a must be even.

Since a and b are co-prime this means that b must be odd.

So 2*2*b^3 has two 2\'s in its prime factorization, and

since a is even we know that a^3 must have at least three 2\'s

in its prime factorization. This gives us a contradiction,

because the prime factorization of a positive integer is unique.

Therefore 4^(1/3) cannot be expressed as a rational number,

i.e., it is irrational.