Short answers, pulled from the story.
Archimedes invented the concept of exponentiation in the 3rd century BCE. He wrote the treatise The Sand Reckoner to express numbers so vast that they could not be written in the standard Greek numeral system. His work formalized the law of exponents using powers of 10 to describe the universe.
René Descartes introduced the modern notation for exponentiation in 1637. He published this innovation in his text La Géométrie. The convention places the exponent as a superscript to the right of the base and has remained unchanged for nearly four centuries.
A number raised to the power of zero is always one. This rule is derived from the empty product convention which states that the product of no numbers is one. The case of zero to the power of zero remains controversial and is generally considered undefined.
Konrad Zuse introduced floating-point arithmetic in his computer Z1 in 1938. This computer used a register to represent the leading digits and another to represent the exponent of 10. This innovation allowed for the representation of numbers of varying magnitudes for scientific and engineering calculations.
The complex numbers that satisfy the equation z to the power of n equals one are called the nth roots of unity. These numbers lie on the unit circle in the complex plane and form the vertices of a regular n-gon. They are fundamental to the study of discrete Fourier transforms and algebraic equations.