Exponent calculator
Raising to the power 0.5 is the same as taking a square root, and to 1/3 is a cube root. It follows from the rule that multiplying powers adds the exponents: x^0.5 × x^0.5 = x^1, so x^0.5 must be the thing that gives x when multiplied by itself. Negative exponents follow the same logic in reverse: x^−1 is 1/x, because x^1 × x^−1 = x^0 = 1.
An exponent is repeated multiplication: 2^10 is 1,024. A negative exponent gives the reciprocal, so 2^−1 is 0.5. A fractional exponent is a root, so 9^0.5 is 3.
How to use exponents
Three rules cover almost all exponent arithmetic. Multiplying powers of the same base adds the exponents. Dividing subtracts them. Raising a power to a power multiplies them. Everything else. Why anything to the power zero is one, why negative exponents are reciprocals, why fractional exponents are roots. Falls out of those three rules being applied consistently rather than being separate facts to memorise.
Questions
1,024 — which is why a kilobyte is 1,024 bytes rather than 1,000.
The reciprocal. 2^−3 is 1 ÷ 2³ = 0.125.
One, for any non-zero base. It follows from dividing a power by itself.
A root. x^(1/2) is the square root, x^(1/3) the cube root.
Because two consistent rules disagree there: anything to the zero is one, and zero to anything is zero. Different contexts choose differently.