# show that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers

Show that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers.

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#### Solution Preview

We will prove this by contradiction.

The idea of a proof by contradiction is to:

First, we assume that the opposite of what we wish to prove is true.

Then, we show that the logical consequences of the assumption include a contradiction.

Finally, we conclude that the assumption must have been false

Recall, A number r is rational if it can be written as a fraction r = p/q where both p and q are integers.

Proof: For 2^1/3

(If 2^1/3 is rational, it should be representable as a fraction r = p/q)

Assume that a rational number r exists such that r^3 = 2. (or, r = 2^1/3)

As we have already assumed that, r is rational, in the representation r=p/q assume p and q are mutually prime, i.e. have no common divisors. The fraction p/q in this case is called irreducible. In other words, a/b is ...

#### Solution Summary

It is shown that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers.