## ::Exponentiation

### ::concepts

Nowrap::''x''    ''n''::''b''    Number::complex    Power::positive    Numbers::integer    Function::''i''

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\$B=1/2}} (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:
$b^n = \underbrace{b \times \cdots \times b}_n$

In that case, bn is called the n-th power of b, or b raised to the power n.

The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b2) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b3) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b.

When n is a negative integer and b is not zero, bn is naturally defined as 1/bn, preserving the property bn × bm = bn + m.

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

Exponentiation is used extensively in many fields, 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.

Calculation results{{#invoke:Navbar|navbar}}
$\scriptstyle\left.\begin{matrix}\scriptstyle\text{summand}+\text{summand}\\\scriptstyle\text{augend}+\text{addend}\end{matrix}\right\}=$ $\scriptstyle\text{sum}$
Subtraction (−)
$\scriptstyle\text{minuend}-\text{subtrahend}=$ $\scriptstyle\text{difference}$
Multiplication (×)
$\scriptstyle\left.\begin{matrix}\scriptstyle\text{factor}\times\text{factor}\\\scriptstyle\text{multiplier}\times\text{multiplicand}\end{matrix}\right\}=$ $\scriptstyle\text{product}$
Division (÷)
$\scriptstyle\left.\begin{matrix}\scriptstyle\frac{\scriptstyle\text{dividend}}{\scriptstyle\text{divisor}}\\\scriptstyle\frac{\scriptstyle\text{numerator}}{\scriptstyle\text{denominator}}\end{matrix}\right\}=$ $\scriptstyle\text{quotient}$
Modulation (mod)
$\scriptstyle\text{dividend}\bmod\text{divisor}=$ $\scriptstyle\text{remainder}$
Exponentiation
$\scriptstyle\text{base}^\text{exponent}=$ $\scriptstyle\text{power}$
nth root (√)
$\scriptstyle\sqrt[\text{degree}]{\scriptstyle\text{radicand}}=$ $\scriptstyle\text{root}$
Logarithm (log)
$\scriptstyle\log_\text{base}(\text{antilogarithm})=$ $\scriptstyle\text{logarithm}$

Exponentiation sections
Intro   History of the notation   Terminology  Integer exponents  Rational exponents  Real exponents  Complex exponents with positive real bases  Powers of complex numbers  Generalizations  Repeated exponentiation  Zero to the power of zero  Limits of powers  Efficient computation with integer exponents  Exponential notation for function names  In programming languages  List of whole-number exponentials  See also  References  External links

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