Mathematical tables: power rules
Tables show common formulas and properties related to exponentiation (power function).

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Exponentiation: general formulas

NameFormula
Power with natural exponentShow sourcean=aaaana^n = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n}
Power with rational exponentShow sourceapq=apq=(aq)pa^{\frac{p}{q}} = \sqrt[q]{a^p} = \left(\sqrt[q]{a}\right)^p
Power with negative exponentShow sourcean=1ana^{-n} = \frac{1}{a^n}
Power to root conversionShow sourcea1n=ana^{\frac{1}{n}} = \sqrt[n]{a}
Multiplication of powers with common baseShow sourceanam=an+ma^n \cdot a^m = a^{n + m}
Multiplication of powers with common exponentShow sourceanbn=(ab)na^n \cdot b^n = \left(a \cdot b\right)^n
Division of powers with common baseShow sourceap:aq=apaq=apqa^p : a^q = \frac{a^p}{a^q} = a^{p - q}
Division of powers with common exponentShow sourcean:bn=anbn=(ab)na^n : b^n = \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n
Power of powerShow source(an)m=anm\left(a^n\right)^m = a^{n \cdot m}

Exponentiation: common exponents

NameFormula
Inverse cubeShow sourcea3=1a3a^{-3} = \frac{1}{a^3}
Inverse squareShow sourcea2=1a2a^{-2} = \frac{1}{a^2}
Number inverse in exponentiation formShow sourcea1=1aa^{-1} = \frac{1}{a}
Fraction inverse in exponentiation formShow source(ab)1=ba\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}
Any number to power zeroShow sourcea0=1a^0 = 1
Square root in exponentiation formShow sourcea12=a0.5=aa^{\frac{1}{2}} = a^{0.5} = \sqrt{a}
Cubic root in exponentiation formShow sourcea13=a3a^{\frac{1}{3}} = \sqrt[3]{a}
Any number to first powerShow sourcea1=aa^1 = a
Square of the number (raise to second power)Show sourcea2=aaa^2 = a \cdot a
Cube of the number (raise to third power)Show sourcea3=aaaa^3 = a \cdot a \cdot a

Exponentiation: one rules

NameFormula
Number inverse in exponentiation formShow sourcea1=1aa^{-1} = \frac{1}{a}
Fraction inverse in exponentiation formShow source(ab)1=ba\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}
Any number to first powerShow sourcea1=aa^1 = a
One to any powerShow source1n=11^n = 1

Exponentiation: power and root relation

NameFormula
Power to root conversionShow sourcea1n=ana^{\frac{1}{n}} = \sqrt[n]{a}
Square root in exponentiation formShow sourcea12=a0.5=aa^{\frac{1}{2}} = a^{0.5} = \sqrt{a}
Cubic root in exponentiation formShow sourcea13=a3a^{\frac{1}{3}} = \sqrt[3]{a}

Exponentiation: zero rules

NameFormula
Any number to power zeroShow sourcea0=1a^0 = 1
Zero to any powerShow source0n=00^n = 0

Some facts

  • Exponentiation means multiplying the same number by itself many times:
    an=aaaana^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: 23=222=82^3 = 2 \cdot 2 \cdot 2 = 8
  • We read symbol 232^3 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, 232^3 is a different number than 323^2.
    ⚠ WARNING! annaa^n \ne n^a
  • Raising any number to the first power does not change the value. For example, 313^1 is 3:
    ⓘ Remember: a1=aa^1 = a
  • In turn raising to zero power gives number one e.g. 303^0 gives 1:
    ⓘ Remember: a0=1a^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. x1x^{-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: (ab)n=(ba)n\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: apq=(aq)p=apqa^{\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:
    0n=0000n=00^n = \underbrace{0 \cdot 0 \cdot 0 \cdot \ldots \cdot 0}_{n} = 0
    1n=1111n=11^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:
    (a)n={anif n is evenanif n is odd \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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