Billy is trying to convince Sam that all square roots are irrational. To support his argument, Billy points out that sqrt(2), sqrt(6), sqrt(7), and sqrt(15) are all irrational. Since this set of numbers contains the square roots of both prime and composite numbers as well as even and odd numbers, Billy argues that there is no number (prime, composite, even, or odd) whose square root is rational. Thus, all square roots must be irrational.

Where has Billy made his error?

Question

Billy is trying to convince Sam that all square roots are irrational. To support his argument, Billy points out that sqrt(2), sqrt(6), sqrt(7), and sqrt(15) are all irrational. Since this set of numbers contains the square roots of both prime and composite numbers as well as even and odd numbers, Billy argues that there is no number (prime, composite, even, or odd) whose square root is rational. Thus, all square roots must be irrational.

Where has Billy made his error?

Vocaloid · Answer

whether a number is even/odd/prime/composite does not determine whether its square root is rational. for a square root to be rational, the radicand must be a perfect square, of which there are many examples (ex. sqrt(4) = 2)