A compact method of writing `5*5*5*5` is `5^4`. The expression `5^4` is written in exponential notation. Similarly, we can write

`(2x)/3*(2x)/3*(2x)/3 \ \ \ \ \ ` as ` \ \ \ \ \ ((2x)/3)^3`

Exponential notation can be used to express the product of any expression that is used repeatedly as a factor.

Definition of Natural Number Exponents

If `b` is any real number and `n` is any natural number, then `b^n=\underbrace{b*b*b* \ cdots \ * b}_\text{n factors of b}` In the expression `b^n`, `b` is the base, `n` is the exponent, and `b^n` is the `n`th power of `b`

For instance,

`(-5)^4=(-5)(-5)(-5)(-5)=625`
`-5^4=-(5*5*5*5)=-625`
Notice the difference between `(-5)^4=625` and `-5^4=-625`. The parentheses in `(-5)^4` indicate that the base is `-5`; however, the expression `-5^4` means `-(5)^4`. This time the base is `5`.

Definition of `b^0`

For any nonzero real number `b`, `b^0=1`.

Any nonzero real number raised to the zero power equals 1. For example,

`7^0=1 \ \ \ \ \ \ \ \ \ \ (1/2)^0=1 \ \ \ \ \ \ \ \ \ \ (-3)^0=1 \ \ \ \ \ \ \ \ \ \ pi^0=1 \ \ \ \ \ \ \ \ \ \ (a^2=1)^0=1`

Definition of `b^(-n)`

If `b!=0` and `n` is any natural number, then `b^(-n)=1/(b^n)` and `1/(b^(-n))=b^n`

Here are some examples of this definition.

`3^(-2)=1/(3^2)=1/9 \ \ \ \ \ \ \ \ \ \ 1/(4^(-3))=4^3=64 \ \ \ \ \ \ \ \ \ \ (5^(-2))/(7^(-1))=(7^1)/(5^2)=7/25`

Restriction Agreement

The expressions `0^0`, `0^n` where `n` is a negative integer, and `x/0` are all undefined expressions. Therefore, all values of variables in this text are restricted to avoid any one of these expressions.

For instance, in the following expression

`(x^0 y^(-3))/(z-4)`
we assume that `x!=0`, `y!=0`, and `z!=4`.

Properties of Exponents

If `m`, `n`, and `p` are integers and `a` and `b` are real numbers, then

Product `\ \ \ \ \ \ \ \ \ \ b^m*b^n=b^(m+n)`

Quotient `\ \ \ \ \ \ \ \ \ \ (b^m)/(b^n)=b^(m-n)`, `\ \ \ \ \ b!=0`

Power `\ \ \ \ \ \ \ \ \ \ (b^m)^n=b^(mn) \ \ \ \ \ (a^mb^n)^p=a^(mp)b^(np) \ \ \ \ \ ((a^m)/(b^n))^p=(a^(mp))/(b^(np))`, `\ \ \ \ \ b!=0`

To simplify an expression involving exponents, write the expression in a form in which each base appears at most once and no powers of powers or negative exponents appear.