To this point, the expression `b^n` has been defined for real numbers `b` and integers `n`. Now we wish to extend the definition of exponents to include rational numbers so that expressions such as `2^(1//2)` will be meaningful. Not just any definition will do. We want a definition of rational exponents for which the properties of integer exponents are true. The following example shows the direction we can take to accomplish our goal.

If the product property for exponential expressions is to hold for rational exponents, then for rational numbers `p` and `q`, `b^pb^q=b^(p+q)`. For example,

`9^(1//2)*9^(1//2)` must equal `9^(1//2+1//2)=9^1=9`
Thus `9^(1//2)` must be the square root of `9`. That is `9^(1//2)=3`.

The example suggests that `b^(1//n)` can be defined in terms of roots according to the following definition.

Definition of `b^(1//n)`

If `n` is an even positive integer and `B >=0`, then `b^(1//n)` is the nonnegative real number such that `(b^(1//n))^n=b`.

As examples,

`25^(1//2)=5` because `5^2=25`
`(-64)^(1//3)=-4` because `(-4)^3=-64`
`16^(1//2)=4` because `4^2=16`
`-16^(1//2)=-(16^(1//2))=-4`
`(-16)^(1//2)` is not a real number.
`(-32)^(1//5)=-2` because `(-2)^5=-32`

If `n` is an even positive integer and `b < 0`, then `b^(1//n)` is a complex number. Complex numbers are discussed in Chapter 1.

To define expressions such as `8^(2//3)`. we will extend our definition of exponents even further. Because we want the power property `(b^p)^q=b^(pq)` to be true for rational exponents also, we must have `(b^(1//n))^m=b^(m//n)`. With this in mind, we make the following definition.

Definition of `b^(m//n)`

For all positive integers `m` and `n` such that `m//n` is in simplest form, and for all real numbers `b` for which `b^(1//n)` is a real number, `b^(m//n)=(b^(1//n))^m=(b^m)^(1//n)`

Because `b^(m//n)` is defined as `(b^(1//n))^m` and `(b^m)^(1//n)`, we can evaluate expressions such as `8^(4//3)` in more than one way. For example, because `8^(1//3)` is a real number, `8^(4//3)` can be evaluated in either of the following ways:

`8^(4//3)=(8^(1//3))^4=2^4=16`
`8^(4//3)=(8^4)^(1//3)=4096^(1//3)=16`

Of the two methods, the `b^(m/n)=(b^(1//3))^m` method is usually easier to apply, provided you can evaluate `b^(1//n)`.

Properties of Rational Exponents

If `p`, `q`, and `r` represent rational numbers and `a` and `b` are positive real numbers, then

Product `\ \ \ \ \ \ \ \ \ \ b^p*b^q=b^(p+q)`

Quotient `\ \ \ \ \ \ \ \ \ \ (b^p)/(b^q)=b^(p-q)`

Power `\ \ \ \ \ \ \ \ \ \ (b^p)^q=b^(pq) \ \ \ \ \ (a^pb^q)^r=a^(pr)b^(qr) \ \ \ \ \ ((a^p)/(b^q))^r=(a^(pr))/(b^(qr)) \ \ \ \ \ b^(-p)=1/(b^p)`