Operation that associates an ordered pair of integers (

*a*,*b*) with the number*a*, called “^{b}*a*raised to the power of*b*“.Just as multiplication corresponds to repeated addition, exponentiation corresponds to repeated multiplication.

- If
*b*is a positive integer, the exponentiation operation indicates that the number*a*is used*b*times as a factor.

Example : \({5}^{4}\) = 5 × 5 × 5 × 5 - If
*b*is a negative integer, the exponentiation operation indicates that the inverse of the number*a*is used*b*times as a factor.

Example : \({5}^{-4}\) = 1/5 × 1/5 × 1/5 × 1/5 - If
*b*= 1, then : \({a}^{b}\) =*a.*Example : \({5}^{1}\) = 5

- If
*b*= 0, then : \({a}^{b}\) = 1*.*Example : \({5}^{0}\) = 1

- If
*a*= 1, then : \({a}^{b}\) = 1*.*Example : \({1}^{4}\) = 1

### Properties

- For every number
*a*, we have :*a*^{1}=*a* - For every number
*a*, we have :*a*^{0}= 1. - For every number
*a*, we have : 1^{a}= 1. - For every non-zero number
*a*, we have : 0= 0.^{a}

### Example

In the expression \({3}^{4}\)= 81, the number 3 is the base, the number 4 is the exponent and the number 81 is the power.

### Educational note

- When
*b*= 0, it can be shown that when*a*is non-zero, the result is 1. - When
*a*= 0, it is more difficult to show that the result is still 1 at primary or secondary levels. Instead, the result is said to be generalized,*by convention*. However, there are natural proofs of this result, using set theory, for example, where the exponentiation operation corresponds to the cardinal of the set of the applications of a set A to a set B :*n*(B^{A}) =*n*(B)^{n(A)}=*n*({*f*|*f*is an application of A to B}).