Prove that the fourth root of 2 is irrational.

To prove that the fourth root of 2 is irrational, we can prove that all
(integer) roots of 2 are

irrational.

Suppose there is a rational root. Any rational number can be written as a/b,
where a and b are

whole numbers, and “a and b have no more common factors”.

If that's true, then the nth root of 2 = a/b. So
, which implies,
. Now if

, it follows that an is even number (since 2 time anything is always even, if
the "anything"

is a whole number).

Notice that if is even, then a must be even.
If a were odd, then no matter how many times you

multiplied it by itself you'd always end up with an odd number.

If a is an even number, we could write it as 2k (k can be anything - the point
is, 2*k must be

even, so writing a like this shows that it is even).

So we have . Divide both sides by 2:

Which shows that (something), so
is even. Recall from earlier, if
is even, then b

must be even.

We have just shown that a and b are both even. But that goes against the earlier
statment that a

and b had no common factors. So we've ended up proving that a/b have no common
factors, and

also that they do have a common factor, which is impossible.

Since the other assumption was "the nth root of 2 is rational", that's another
way of saying "it's

impossible that the nth root of 2 is rational" or "the nth root of 2 is always
irrational"

QED