Question

Proove that:
a)2^(1/2) is not rational.
b) prove that 2^(1/3) is not rational.

I have managed to prove the first part, but i am stuck on the second can someone give me an hint please?

Answer 1

assume 2^(1/2) is a rational number in the form a/b, where a/b is reduced to lowest terms

2^(1/3) = a/b
cube both sides
2= a^3/b^3
2b^2 = a^3
the left side of this equation is even
then the right side of the equation must be even
Properties of integers:
if you cube an even number the result is even
so a must be even.
Which means we could write a = 2k
and back in 2b^3 = a^3 , we can say
2b^3 = 8k^3
b^3= 4k^3
Now the right side is even, which means the LS is even and b itself must be even

so our original assumption that 2^(1/3) = a/b , where a/b is in lowest terms, led us to conclude that both a and b were even.
This is a contradiction, since a/b can be further reduced.
Thus our assumption that 2^(1/3) is rational MUST BE FALSE
Therefore 2^(1/3) is NOT rational

Note that this follows almost exactly what you must have done for the first question