negative exponent

A negative exponent means “take the reciprocal”: b^(−n) = 1 / b^n, for a nonzero base b. It does not make the number negative; it flips it into a fraction. So 2^(−3) is not −8 but 1/8.

The reason follows the pattern of powers. Going up multiplies by the base, so going down divides by it: 2^3 = 8, 2^2 = 4, 2^1 = 2, 2^0 = 1, 2^(−1) = 1/2, 2^(−2) = 1/4. Each downward step keeps dividing by 2, which is exactly what the reciprocal rule encodes. The quotient rule b^m/b^n = b^(m−n) also forces this when m < n.

A handy consequence is that a factor can move between numerator and denominator by changing the sign of its exponent: 1 / b^(−n) = b^n. Two common slips: thinking a negative exponent makes a value negative, and forgetting that the base must be nonzero (since 0 has no reciprocal).