Exponent Calculator
Any base, any exponent, including the ones that look strange.
Exponent Calculator
What the Exponent Counts
An exponent counts how many times the base is multiplied by itself. Two to the fifth is 2 times 2 times 2 times 2 times 2, which is 32. That plain reading covers positive whole numbers and nothing else, which is where the confusion usually starts.
Negative and Fractional Exponents
A negative exponent means a reciprocal, not a negative answer. Two to the minus three is one over two cubed, which is one eighth, and is firmly positive.
A fractional exponent is a root. Nine to the power of one half is the square root of nine, which is 3. Eight to the power of one third is the cube root of eight, which is 2. Combining the two, 8 to the power of minus one third is one half.
Why Anything to the Zero Is 1
Dividing powers subtracts exponents, so 2 to the fifth divided by 2 to the fifth is 2 to the zero. Any number divided by itself is 1, so 2 to the zero must be 1. The same reasoning holds for every base except zero itself, where it is left undefined.
The Laws Worth Memorising
Multiplying powers of the same base adds exponents, dividing subtracts them, and a power raised to a power multiplies them. Those three cover most of what an exam will ask, and they are the reason index questions can usually be simplified before anything is calculated.