A number raised to a power, its square root and its nth root.
A negative exponent means the reciprocal: a⁻ⁿ is 1 / aⁿ. So 2⁻³ is 1/8 = 0.125. The result is never a negative number — the minus applies to the exponent, not the base. That mix-up is one of the most common when powers are written by hand.
It follows from the division rule: aⁿ / aⁿ is 1, and by the rule that same expression is aⁿ⁻ⁿ, or a⁰. So a⁰ can only be 1. The single exception is 0⁰, which is defined differently in different contexts or left undefined.
A root is a power with a fractional exponent: √a is a^(1/2), and the nth root is a^(1/n). So the fourth root of 16 is 16^0.25 = 2. Writing them this way lets every exponent rule apply to roots, and it is how this calculator computes them.
Among the real numbers no square is negative — squaring either a positive or a negative gives a positive. So √−9 does not exist in the reals. Odd roots are fine, though: the cube root of −8 is −2, because −2 cubed really is −8.