Exponentiation Explained

Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication:

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: a to the n-th power or a to the power [of] n, or even more briefly: 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, at least 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 e^z. 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.

Exponentiation with 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 expressiona³ = 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.

Exponents one and zero

Notice that 3¹ is the product of only one 3, which is evidently 3.

Also note that 3⁵ = 3·3⁴. Also 3⁴ = 3·3³. Continuing this trend, we should have

3¹ = 3·3⁰.Another way of saying this is that when n, m, and n − m are positive (and if x is not equal to zero), one can see by counting the number of occurrences of x that

Extended to the case that n and m are equal, the equation would read since both the numerator and the denominator are equal. Therefore we take this as the definition of x⁰.

Therefore we define 3⁰ = 1 so that the above equality holds. This leads to the following rule:

• Any number to the power 1 is itself.
• Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0⁰ is discussed below.

Combinatorial interpretation

For nonnegative integers n and m, the power n^m equals the cardinality of the set of m-tuples from an n-element set, or the number of m-letter words from an n-letter alphabet.

0⁵ = |