Question

Describing a Value with a Negative Base
If a base is negative, how will you know whether the answer is positive or negative before evaluating it?

Answer 1

Use the parity of the exponent (and the kind of exponent) to decide the sign.

- Integer exponents:
- (-a)^n with a>0: if n is even the value is positive, if n is odd the value is negative.
- Reason: (-a)^n = (-1)^n · a^n, and (-1)^n = 1 for even n, = −1 for odd n.
- n = 0: (-a)^0 = 1 (except 0^0 is undefined).
- Negative integer n: (-a)^{−m} = 1/((-a)^m), so the sign follows the parity of m.

- Rational exponents p/q in lowest terms:
- If q is odd you can take an odd root of a negative number and get a real result; sign follows the numerator parity:
e.g. (-8)^{1/3} = −2.
- If q is even, the real root does not exist (the value is nonreal/complex).

- General real (nonrational) exponents:
- A negative base raised to a non-integer real exponent is generally complex (use complex logarithm: (-a)^x = a^x e^{iπx}).

Examples:
- (-2)^4 = +16 (even exponent).
- (-2)^3 = −8 (odd exponent).
- (-2)^{−3} = −1/8 (odd numerator → negative).
- (-8)^{2/3} = ((−8)^{1/3})^2 = (−2)^2 = 4 (denominator 3 odd → real).
- (-8)^{1/2} is not a real number.

So: for integer exponents check parity; for rational exponents check denominator odd/even; for arbitrary real exponents expect complex values.