# exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a:

just as multiplication by a positive integer corresponds to repeated addition:

The exponent is usually shown as a superscript to the right of the base. The exponentiation an can be read as: a raised to the n-th power, a raised to the power [of] n, or possibly a raised to the exponent [of] n, or more briefly as a to the n. Some exponents have their own pronunciation: for example, a2 is usually read as a squared and a3 as a cubed.

The power an can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists, but when the base a is a positive real number, an can be defined for all real and even complex exponents n via the exponential function ez. Trigonometric functions can be expressed in terms of complex exponentiation.

Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.

Exponentiation is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

## Terminology

When this article refers to 'an odd power' of a number it means the exponent is an odd number, not that the result is odd. For instance 23 which is 8 is an odd power of 2 because the exponent is 3. This is the usual usage and applies to any similar form like an even power, negative power, or positive power.

## Integer exponents

The exponentiation operation with integer exponents requires only elementary algebra.

### Positive integer exponents

The expression a2 = a·a is called the square of a because the area of a square with side-length a is a2.

The expression a3 = a·a·a is called the cube, because the volume of a cube with side-length a is a3.

So 32 is pronounced "three squared", and 23 is "two cubed".

The exponent says how many copies of the base are multiplied together. For example, 35 = 3·3·3·3·3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, the fifth power of 3, 3 raised to the fifth power, or 3 to the power of 5.

The word "raised" is usually omitted, and very often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five".

Formally, powers with positive integer exponents may be defined by the initial condition a1 = a and the recurrence relation an+1 = a·an.