Consider the following statement: For nil real numbers r, if r3 is irrational th

Question

Consider the following statement: For nil real numbers r, if r3 is irrational then r is irrational. Prove the statement either by contradiction or by contraposition. Clearly indicate which method you are using. If you used proof by contradiction in part (a), write what you would "suppose" and what you would "show" to prove the statement by contraposition. If you used proof by contraposition in part (a), write what you would "suppose" and what you would "show" to prove the statement by contradiction.

Explanation / Answer

By Contradiction. Suppose: there exists an irrational number a, whose cubic root b = a^1/3 is rational. We need to show: that is b is rational, a must not be rational. Since b is rational, then b*b*b is rational. But b*b*b = b^3 = a. It contradicts with the assumption that a is irrational. This proves the claim by contradiction.

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