Tables show common formulas and properties related to exponentiation (power function).

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Exponentiation: general formulas#

Name	Formula
Power with natural exponent	Show source $a^n = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n}$
Power with rational exponent	Show source $a^{\frac{p}{q}} = \sqrt[q]{a^p} = \left(\sqrt[q]{a}\right)^p$
Power with negative exponent	Show source $a^{-n} = \frac{1}{a^n}$
Power to root conversion	Show source $a^{\frac{1}{n}} = \sqrt[n]{a}$
Multiplication of powers with common base	Show source $a^n \cdot a^m = a^{n + m}$
Multiplication of powers with common exponent	Show source $a^n \cdot b^n = \left(a \cdot b\right)^n$
Division of powers with common base	Show source $a^p : a^q = \frac{a^p}{a^q} = a^{p - q}$
Division of powers with common exponent	Show source $a^n : b^n = \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$
Power of power	Show source $\left(a^n\right)^m = a^{n \cdot m}$

Exponentiation: common exponents#

Name	Formula
Inverse cube	Show source $a^{-3} = \frac{1}{a^3}$
Inverse square	Show source $a^{-2} = \frac{1}{a^2}$
Number inverse in exponentiation form	Show source $a^{-1} = \frac{1}{a}$
Fraction inverse in exponentiation form	Show source $\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}$
Any number to power zero	Show source $a^0 = 1$
Square root in exponentiation form	Show source $a^{\frac{1}{2}} = a^{0.5} = \sqrt{a}$
Cubic root in exponentiation form	Show source $a^{\frac{1}{3}} = \sqrt[3]{a}$
Any number to first power	Show source $a^1 = a$
Square of the number (raise to second power)	Show source $a^2 = a \cdot a$
Cube of the number (raise to third power)	Show source $a^3 = a \cdot a \cdot a$

Exponentiation: one rules#

Name	Formula
Number inverse in exponentiation form	Show source $a^{-1} = \frac{1}{a}$
Fraction inverse in exponentiation form	Show source $\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}$
Any number to first power	Show source $a^1 = a$
One to any power	Show source $1^n = 1$

Exponentiation: power and root relation#

Name	Formula
Power to root conversion	Show source $a^{\frac{1}{n}} = \sqrt[n]{a}$
Square root in exponentiation form	Show source $a^{\frac{1}{2}} = a^{0.5} = \sqrt{a}$
Cubic root in exponentiation form	Show source $a^{\frac{1}{3}} = \sqrt[3]{a}$

Exponentiation: zero rules#

Name	Formula
Any number to power zero	Show source $a^0 = 1$
Zero to any power	Show source $0^n = 0$

Some facts#

Exponentiation means multiplying the same number by itself many times:

$a^n = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n}$
where:
- a - the base of exponentiation, it's a number which we multiply by itself,
- n - the exponent of exponentiation, it's a number of multiplications performed.
ⓘ Example: $2^3 = 2 \cdot 2 \cdot 2 = 8$
We read symbol $2^3$ as "two to the third power" or more colloquially: "two to third".
Formally, the exponentiation is a two-argument operation, where the first argument is the base (number 2 in above example) and the second one is exponent (number 3 in the above example).
Exponentiation is not commutable, i.e. you can not swap the base with the exponent. For example, $2^3$ is a different number than $3^2$ .

⚠ WARNING! $a^n \ne n^a$
Raising any number to the first power does not change the value. For example, $3^1$ is 3:

ⓘ Remember: $a^1 = a$
In turn raising to zero power gives number one e.g. $3^0$ gives 1:

ⓘ Remember: $a^0 = 1$
Exponentiation by the negative number is the same as performing an identical operation, but with inverse base. Therefore, often the inverse operation is written as an increase to the power of -1, e.g. $x^{-1}$ means as much as "the inverse of x". If you want to know more about inverse of the numbers, then you can check out our other calculator: Fractions: inverse (reciprocal). In general, the folowing formula is met

ⓘ Remember: $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$
Exponentiation with a non-integer exponent is the same as root. The exponentiation with rational exponent may be used to present root and power in one operation. In general, the following formula is met:

ⓘ Remember: $a^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}$
Basing on above formula we can see that the power and root are in fact the same kind of operation. In practice, this means that it does not matter in what order we perform these operation (we can calculate root first and raise to the power next or vice versa). In both cases we will get the same result.
Raising the number one to any power gives one. Similarly, zero to any power gives zero. These facts result from property of multiplication by one and zero:

$0^n = \underbrace{0 \cdot 0 \cdot 0 \cdot \ldots \cdot 0}_{n} = 0$

$1^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{n} = 1$
If you want to learn more about elementary math operations such as multiplication you can check out our other calculator: Number operations.
If the exponentiation base is negative, then the sign of the result depends on parity of exponent. Even exponents give positive result and odd exponents give the negative one. In general, we can write:

$\left(-a\right)^n = \left\{ \begin{array}{ll} a^n & \textrm{if n is even}\\ -a^n & \textrm{if n is odd}\\ \end{array} \right.$

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