Raising to a power began as repeated multiplication
Exponentiation pairs a base b with an exponent n. For positive integers, b^n means multiplying b by itself n times; the idea then stretches to zero, negatives, reals, complex numbers, and even matrices. Its simple rules, such as adding exponents when multiplying powers, explain why anything raised to zero is 1.
Exponentiation takes two numbers, a base b and an exponent n, the second usually written as a superscript to the right of the base, or with a caret or up arrow in plain text. When n is a positive integer it is nothing more than repeated multiplication: b times itself n times. The exponent is also called the power, or in British English the index, and the result is likewise called a power. Aloud, b^n becomes 'b to the power n', 'the nth power of b', or simply 'b to the n'.
Two everyday names come from geometry. A square of side b has area b², so b² is 'b squared'; a cube of side b has volume b³, hence 'b cubed'. Because multiplication is associative, multiplying two powers of the same base adds their exponents, b^m · b^n = b^(m+n), and raising a power to a further power multiplies them, (b^m)^n = b^(mn).
Those rules dictate the less obvious cases. Any number to the power 0 is generally defined as 1, which keeps b^0 · b^n = b^n true and agrees with the convention that an empty product equals 1, though zero to the power zero can be an exception depending on context. Negative exponents mean reciprocals: for nonzero b, b^(−n) equals 1 divided by b^n. A more involved construction admits arbitrary complex numbers as both base and exponent, and other generalizations let objects such as square matrices serve as the base.
In abstract algebra, where more than one operation could be repeated, writers show which one by placing its symbol in the superscript ahead of the exponent. For a function f, f^n(x) can mean the value f(x) multiplied by itself n times, while f^∘n(x) means feeding f its own output n times over. The operation is basic equipment throughout mathematics, science, and engineering.
Source: Exponentiation
