The readers may find that literatures on mathematics provide various definitions of powers. Some authors do not distinguish the term power from index/exponent. Throughout this article, a * power* is a mathematical expression in the form of

*b*where . The number

^{r}*r*is called an

*or*

**index***, and*

**exponent***b*is called a

*. By the definition,*

**base***b*is a power with the base

^{r}*b*and exponent

*r*.

**Case 1: r is a natural number**

Below are how *b ^{r}* are defined for r = 1, 2, 3, …

*b*^{1} = *b* ;

*b ^{r}* =

*b*

_{1 }⋅

*b*

_{2 ⋅ }

*b*

_{3}⋅ … ⋅

*b*with

_{r}*b*

_{1}=

*b*

_{2}=

*b*

_{3}= … =

*b*=

_{r}*b*;

*r*= 2, 3, 4, …

**Case 2: r = 0**

*b*^{0} = 1 ; *b* ≠ 0

Note: 0^{0} is not defined!

**Example 1**

5^{3} = 5⋅5⋅5 = 125

(0.1)^{4} = 0.1 ⋅ 0.1 ⋅ 0.1 ⋅ 0.1 = 0.0001

**Case 3: r is a negative integer**

; , *b* ≠ 0.

Note: is the set of all natural numbers.

**Example 2**

**Case 4: r is a rational number**

; , *b* ≥ 0

Thus, *b*^{½} is the square root of *b*.

**Example 3**

; , *b* ≥ 0

**Example 4**

;

**Example 5**

; , *b* ≥ 0.

**Example 6**

;

**Example 7**

; , *b* > 0

**Example 8**

; , *b* ≠ 0

**Example 9**

How if the exponents are irrational numbers? For example, how to compute ? Indeed, the number is irrational as well, whose decimal representation is an approximation. This will be discussed later on this website.

**Some formulae related to powers**

(Apply the following formulae cautiously, look at the description below R5.)

Let . Then,

R1:

R2: ; *b* ≠ 0

R3:

R4:

R5: ; *b* ≠ 0

Cautions

The formulae R1 through R5 hold, provided that the powers and algebraic operations are defined. For example, we must not apply R1 in 0^{5⋅}⋅0^{-2} = 0^{3} because 0^{-2} is undefined. As another example of misusing R1 is . It is incorrect because is undefined and so are and .

**Exponential functions**

The exponential function *f* is the real-valued function whose domain is and for every *x* ∈ *f*(x) = *e ^{x}*, where

*e*is the Euler number. Below is the graph of the exponential function.

The function has the following properties.

- It is strictly increasing, that is x
_{1}< x_{2}⇒*f*(x_{1}) <*f*(x_{2}). - The coordinates of the y-intercept is (0,1).
- The x-axis is the asymptote to the curve of the function. The graph of the function gets closer and closer to the axis as
*x*gets smaller and smaller. - The formulae R1 through R3 above apply.