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

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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$\left(a^n\right)^m = a^{n \cdot m}$ |

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$a^3 = a \cdot a \cdot a$ |

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$1^n = 1$ |

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$a^{\frac{1}{3}} = \sqrt[3]{a}$ |

Name | Formula |

Any number to power zero | Show source$a^0 = 1$ |

Zero to any power | Show source$0^n = 0$ |

**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 myltiply 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
**zeroth 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: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ - 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 operationtion**. 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 depend on
**parity od exponent**.**Even exponents**gives**positive**result and**odd exponents**gives 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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