Short answers, pulled from the story.
Exponentiation is an arithmetic operation involving two numbers, the base and the exponent. When the exponent is a positive integer, exponentiation means multiplying the base by itself that many times, so three to the fifth power is three multiplied five times.
Michael Stifel coined the word exponent in 1544. The term originates from the Latin exponentem, the present participle of exponere, meaning to put forth.
In The Sand Reckoner, Archimedes proved the law of exponents needed to manipulate powers, then used powers to estimate the number of grains of sand that could fill the universe.
A nonzero number raised to the zero power equals one, a value also produced by the empty product convention, which applies in any algebraic structure whose multiplication has an identity. The case of zero raised to the zero power is controversial and depends on context.
The Diffie, Hellman key exchange uses exponentiation in finite fields for secure communication. Exponentiation is computationally inexpensive, while the inverse operation, the discrete logarithm, is computationally expensive, so an exponent cannot be practically recovered when the field is large enough.
Exponentiation by squaring is an efficient method for computing powers with integer exponents. Computing two to the hundredth power by plain repeated multiplication needs 99 multiplications, but applying Horner's rule to the binary exponent reduces this to 8.
Programming languages express exponentiation as an infix operator or a function. The caret symbol is most common and replaced the ASCII uparrow in 1967, while many languages such as Python, Ruby, and Fortran use a double asterisk.