The law is stated as follows: The power of a quotient is equal to the quotient obtained when the dividend and divisor are each raised to the indicated power separately, before the division is performed.

Thus far in this discussion of exponents, the emphasis has been on exponents which are positive integers. There are two types of exponents which are not positive integers, and two which are treated as special cases even though they may be considered as positive integers.

Zero occurs as an exponent in the answer to a problem such as 4³ + 4³. The law of exponents for division states that the exponents are to be subtracted. This is illustrated as follows:

Another way of expressing the result of dividing 4³ by 4³ is to use the fundamental axiom which states that any number divided by itself is 1. In order for the laws of exponents to hold true in all cases, this must also be true when any number raised to a power is divided by itself. Thus, 4³/4³ must equal 1.

Since 4³/4³ has been shown to be equal to both 4⁰ and 1, we are forced to the conclusion that 4⁰ = 1. 

By the same reasoning,

Thus we see that any number divided by itself results in a 0 exponent and has a value of 1. By definition then, any number (other than zero) raised to the zero power equals 1. This is further illustrated in the following examples:

The number 1 arises as an exponent sometimes as a result of division. In the example 5³/5² we subtract the exponents to get

This problem may be worked another way as follows:

We conclude that any number raised to the first power is the number itself. The exponent 1 usually is not written but is understood to exist.

If the law of exponents for division is extended to include cases where the exponent of the denominator is larger, negative exponents arise. Thus,

Another way of expressing this problem Is as follows:

We conclude that a number N with a negative exponent is equivalent to a fraction having the following form: Its numerator is 1; its denominator is N with a positive exponent whose absolute value is the same as the absolute value of the original exponent. In symbols, this rule may be stated as follows:

The following examples further illustrate the rule:

Notice that the sign of an exponent may be changed by merely moving the expression which contains the exponent to the other position in the fraction. The sign of the exponent is changed as this move is made. For example,