Mathematical tables: power rules
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$a^{\frac{1}{3}} = \sqrt{a}$ 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$a^{\frac{1}{3}} = \sqrt{a}$

# 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: $a^{\dfrac{p}{q}} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}$

• 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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