Exponentiation is a mathematical operation $$a^b = a$$ multiplied by itself $$b$$ times. For example, $$3^3 = 3 \times 3 \times 3 = 27$$. It is ubiquitous in modern mathematics. $$a^b$$ is typically pronounced "$$a$$ to the $$b$$th power" or $$a$$ to the $$b$$th." $$a$$ is called the base, and $$b$$ the exponent. The adjective form of exponentiation is exponential.

In googology, it is the third hyper operator. When repeated, it forms tetration.

In Multiplication, It is: .

In Addition, It is: .

In the fast-growing hierarchy, $$f_2(n) = n \times 2^n$$ corresponds to exponential growth rate.

## Definition

For a real number $$a$$ and a non-negative integer $$b$$, exponentiation has the following definition:

$a^b := \prod_{i = 1}^{b} a$

More precisely, it is defined in the following recursive way:

$a^b := \left\{ \begin{array}{ll} 1 & (b = 0) \\ a^{b-1} \times a & (b > 0) \end{array} \right.$

For a positive real number $$a$$ and a real number $$b$$, $$a^b$$ is defined as $$e^{b \ln a}$$, where $$e^x$$ is the exponential function and $$\ln$$ is the natural logarithm, which are defined like so:

$e^x := \sum_{i = 0}^{\infty} \frac{x^i}{i!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$ $\ln x := \int_1^x \frac{dt}{t}$

Here $$n!$$ denotes the factorial of $$n$$. This allows the definition to be expanded to non-integer exponents. Despite the $$x^i$$ terms in the definition of $$e^x$$, the definition is not circular, because we defined $$a^b$$ for the case where $$b$$ is a non-negative integer in a distinct way. Moreover, the values of $$a^b$$ with respect to those two definitions coincide with each other for any positive real number $$a$$ and any non-negative integer $$b$$.

Similarly, the exponentiation is extended to elements in other complete topological rings such as complex numbers, $$p$$-adic numbers, and matrices with appropriate coefficients. This is a special property of exponentiation, and few analogues are known for hyper operators. At least, the complex extension of tetration is much more difficult than exponentiation.

## Properties of exponentiation

The following are identities of exponentiation:

$a^0 = 1$
$a^1 = a$
$1^a = 1$
$0^a = 0$

Here $$a$$ is a real number, and is non-zero in the fourth equality. Although we set $$0^0 := 1$$ in the previous section in order to define $$e^x$$, $$0^0$$ has different values, e.g. $$0$$ or $$1$$, depending on context; it is often treated as undefined for this purpose.

The following are some useful properties in manipulating exponents:

$a^{-b} = \frac{1}{a^b}$
$a^{b + c} = a^b \cdot a^c$
$a^{b - c} = \frac{a^b}{a^c}$
$a^{b \cdot c} = \left(a^b\right)^c$

Here variables in each equality are real numbers such that both hand sides make sense. For example, in the first equiality, $$a$$ should be non-zero, and should be positive unless $$b$$ is an integer.

These can be proved by expressing the exponents in terms of the exponential function.

For a non-negative real number $$a$$ and a positive integer $$b$$, $$a^{1/b}$$ is often written $$\sqrt[b]{a}$$, called radical notation. When $$b = 2$$, it is usually left out: $$\sqrt{a}$$. This is called the square root of $$a$$.

Unlike the previous two hyper-operators, i.e. addition and multiplication, exponentiation is neither commutative nor associative. For example, $$3^5 = 243 > 125 = 5^3$$, and $$3^{2^3} = 6,561 > 729 = \left(3^2\right)^3$$. Note that when $$a$$ and $$b$$ are integers, $$a^b\ne b^a$$ except when $$a=b$$ or they are 2 and 4.

Repeated exponentiation is solved from right to left. For example, abcd = a(b(cd)).

If the exponentiation is with other operators in the math sentences, the ^ will be solved first like a*b^c = a*(b^c).

## Applications

In calculus

Two important rules of calculus are the Power Rules of Differentiation and Integration:

$\frac{d}{dx}x^n = nx^{n - 1}$

$\int x^n dx = \frac{1}{n + 1}x^{n + 1} + C$

Here $$n$$ is a real number, and satisfies $$n \neq -1$$ in the second equality. The domain of the functions in the right hand sides depends on the value of $$n$$.

### Notations

The exponential function $$a^b$$ can be represented:

• In arrow notation as $$a \uparrow b$$.
• In chained arrow notation as $$a \rightarrow b$$ or $$a \rightarrow b \rightarrow 1$$.
• In BEAF as $$\{a, b\}$$ or $$a\ \{1\}\ b$$.
• In Hyper-E notation as E(a)b.
• In plus notation as $$a +++ b$$.
• In star notation (as used in the Big Psi project) as $$a ** b$$.
• In the programming languages Python and Ruby, it is written as a ** b.

## Special exponents

The case $$a^2$$ is called the square of $$a$$, because it is the area of a square with side length $$a$$. Likewise, $$a^3$$ is the cube of $$a$$. $$a^4$$ is sometimes called the tesseract of $$a$$, but this term is not used frequently.