FANDOM

10,265 Pages

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.

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

In the fast-growing hierarchy, $$f_2(n)$$ corresponds to exponential growth rate.

Definition Edit

For nonnegative integers $$b$$, exponentiation has the following definition:

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

For general values of $$b$$, $$a^b$$ 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:

$e^x := 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$ $\ln x := \int_1^x \frac{dt}{t}$ (where 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.

Properties of exponentiation Edit

The following are identities of exponentiation:

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

$$0^0 = 0 \text{ or } 1$$ has different values depending on context; it is often treated as undefined for this purpose.

The following are some useful properties in manipulating exponents:

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

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

$$a^{1/b}$$ 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, addition and multiplication, exponentiation is neither commutative nor associative. For example, $$3^5 = 243 > 125 = 5^3$$, and $$3^{2^3} = 6,561 > 729 = \left(3^2\right)^3$$.

This sould be noted that $$a^b\ne b^a$$ except when they are both same or they are 2 and 4 when they are integer, though they the hyper operator on the positive integers.

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

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

ApplicationsEdit

In calculus

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

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

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

Notations Edit

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 $$a\ \{1\}\ b$$.[1]
• 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 Edit

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.

1. [1]