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MATH 430 Chapter 1 Exercise 6

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

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