# Exponentiation
**A binary operation involving a base and an exponent which, for positive integer exponents, is equal to repeated multiplication of the base.**
![[Exponentiation.svg|500]]
*Graphs of exponentiations of different bases; every graph intersects the point $(0, 1)$ and $(1, b)$, where $b$ is the base; every graph also never reaches a value of zero*
*Exponentiation* is a binary operation involving two numbers, a *base* $b$ and an *exponent* $n$. This operation is denoted as $b^{n}$ and referred to as "$b$ to the power of $n
quot;, "$b$ raised to the $n$th power", or "$b$ to the $nquot;.
It is defined separately depending on the type of exponent, but in elementary arithmetic it is defined only for positive integer exponents where it is equal to *repeated multiplication* of the base, that is, the product of $n$ bases.
$b^{n}=\underbrace{b\times b\times\dots\times b}_{n\;\text{times}}$
The operations $b^{2}$ and $b^{3}$ are known as "$b$ squared" and "$b$ cubed", respectively, while $b^{-1}=\frac{1}{b}$ is known as the "multiplicative inverse" or "reciprocal" of $b$.
The *[[exponential function]]* refers specifically to the named function $f(x)=e^{x}$ where $e$ is [[e|Euler's number]].
Exponentiation of a base $b$ is the inverse operation of the [[logarithm]] in base $b$.
## Types of exponents
### Integer exponents
> [!NOTE] Exponentiation with positive integer exponents
> As stated above, the definition of exponentiation with *positive* integer exponents is *repeated multiplication* of the base, specifically the product of multiplying $n$ bases together.
> $b^{n}=\underbrace{b\times b\times\dots\times b}_{n\;\text{times}}$
>
> It is formally defined using mathematical induction where the base case is $b^{1}=b$ and the recurrence relation is $b^{n+1}=b^{n}\times b$.
> [!NOTE] Exponentiation with the zero exponent
> Any *non-zero* number raised to the power of zero is $1$.
> $b^{0}=1$
> [!NOTE] Exponentiation with negative integer exponents
> Exponentiation with *negative* integer exponents is defined for any integer $n$ and non-zero $b$ as
> $b^{-n}=\frac{1}{b^{n}}$
The sum of two numbers raised to the power of integer exponents can be calculated from the powers of the numbers using the [[binomial theorem]].
### Rational exponents
> [!NOTE] Exponentiation with rational exponents
> If $b$ is a *positive* real number and $\frac{m}{n}$ is a rational number with $m$ and $n$ being positive integers, then
> $b^{\frac{m}{n}}=\sqrt[n]{b^{m}}$
Exponentiation of negative real bases with rational exponents are handled in the same way as the exponentiation of complex bases with rational exponents.
### Complex exponents with positive real base
> [!NOTE] Exponentiation with complex exponents
> The exponentiation of a *positive real* base $b$ with a *[[Complex Number|complex]]* exponent $z$ is defined in terms of the [[exponential function]], specifically the natural logarithm, defined for complex arguments.
> $b^{z}=e^{z\ln b}$
> - $\ln b$ - the *[[natural logarithm]]* of $b$
### Complex base with rational exponents
The exponentiation of a *complex* base with a *rational* exponent is most commonly defined with the base expressed in [[Complex Number#Polar form|polar form]].
> [!NOTE] Exponentiation of a complex base with an integer exponent
> The exponentiation of a *complex* base $z$ with an *integer* exponent $n$ is
> $z^{n}=r^{n}\angle n\phi$
>
> This derives from the polar form of the [[Complex Number#^e47b34|product]] of two complex numbers.
> [!NOTE] Exponentiation of a complex base with a rational exponent
> The *principal value* of the exponentiation of a complex base $z$ with a rational exponent $\frac{1}{n}$ is
> $z^{\frac{1}{n}}=\sqrt[n]{z}=\sqrt[n]{r}\angle{\frac{\phi}{n}}$
>
> This derives from the polar form of the [[Complex Number#^e3a9f7|division]] of two complex numbers.
As [[List of Trigonometric Identities#Angle shifts|angle shifts]] of $2k\pi$ for some integer $k$ does not change the value of sine and cosine, if $\phi$ is the argument of a complex number, then any angle of the form $\phi+2k\pi$ is also an argument for the same complex number.
> Thus, the exponentiation of a complex base $z$ with a rational exponent has *multiple values*, known as the $n$th roots of $z$, of the form
> $\sqrt[n]{z}=\sqrt[n]{r}\angle\frac{\phi+2k\pi}{n}$
> where $k=0, 1,\dots, n-1$ and there are always exactly $n$ $n$th roots of any complex number. The root with the largest real part and the one with positive imaginary part is chosen as the *principal value* or *principal root*.
The $n$th roots of $1$ are commonly used and are known specially as [[Root of Unity|roots of unity]].
On the complex plane, the $n$th roots of any complex number can be graphed as a set of points evenly spaced around a circle with radius equal to their shared magnitude.
![[ExponentiationRootsOfComplexNumber.svg|600]]
*The five fifth roots of a complex number; the principal root is in dark blue*
## Index laws
The following laws are only applicable for real bases and exponents.
- $b^{m}\times b^{n}=b^{m+n}$
- $b^{m}\div b^{n}=b^{m-n}$
- $(b^{m})^{n}=b^{mn}$
- $b^{-m}=\frac{1}{m}$
- $b^{\frac{m}{n}}=\sqrt[n]{a^{m}}$