# Exponentiation

## Exponentiation

Operation that associates an ordered pair of integers (a, b) with the number ab, called “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 : a1 = a
• For every number a, we have : a0 = 1.
• For every number a, we have : 1a = 1.
• For every non-zero number a, we have : 0a = 0.

### 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(BA) = n(B)n(A) = n({f | f is an application of A to B}).