if -a^(-b^-c) is a positive integer and a, b, and c are integers, then...

Question

(a) a must be negative (b) b must be negative (c) c must be negative (d) b must be an even positive integer (e) none of the above

Answer 1

$$\displaystyle n=-a^{-b^{-c}}=-\left(a^{b^{-c}}\right)^{-1}=-\frac{1}{a^{b^{-c}}}=-\frac{1}{a^{1/b^c}}=-\frac{1}{\sqrt[b^c]{a}}.$$

We can exclude (D) because $b$ being even implies $b^c$ being even, and an even root, if it exists, is always positive, giving a negative $n$.

Similarly, the answer is (A) because for $n$ to be positive, we need a negative denominator, and $\sqrt[y]{x}$ is negative if and only if $x$ is negative.

In particular, provided that $a$ is negative, $n$ can be a positive integer only if:

• $a=-1$ and $b$ is odd
• $a=-1$ and $c=0$
• both $b$ and $c$ are negative and odd, e.g. $\displaystyle -\frac{1}{\sqrt[(-3)^{-3}]{-2}}=134217728 $