Raise any base to any power, including negative and fractional exponents.
An exponent tells you how many times to multiply a number (the base) by itself. Positive integer exponents are straightforward repeated multiplication. A negative exponent means 'take the reciprocal': b^(−n) = 1/bⁿ. A fractional exponent is a root: b^(1/2) is the square root, b^(1/3) the cube root, b^(p/q) is the q-th root of bᵖ. Negative bases with even-denominator fractional exponents produce complex (imaginary) numbers.
Integer exponent
b^n = b × b × … × b (n times)
Negative exponent
b^(−n) = 1 / b^n
Fractional exponent
b^(p/q) = q-th root of b^p
Any non-zero base raised to the power of zero equals 1. The special case 0⁰ is mathematically indeterminate, but by convention most calculators and programming languages return 1.
A negative exponent means 'divide one by the positive power': 2^(−3) = 1 / 2³ = 1/8 = 0.125. It does not make the result negative.
Only when the denominator of the fraction is odd. (−8)^(1/3) = −2 because (−2)³ = −8. (−4)^(1/2) would require the square root of a negative number, which is an imaginary number (2i), not a real number.