Zero Exponent

We have used positive and negative integral exponents, but we have not yet seen the integer 0 used as an exponent. Note that the product rule was stated to hold for any integers m and n. If we use the product rule on 2³ · 2^-3, we get 2³ · 2^-3 = 2⁰.

Because 2³  8 and , we must have 2³ · 2^-3 = 1. So for consistency we define 2⁰ and the zero power of any nonzero number to be 1.

Zero Exponent

If a is any nonzero real number, then a⁰ = 1.

Example 

Using zero as an exponent

Simplify each expression. Write answers with positive exponents and assume all variables represent nonzero real numbers.

Solution

a) To evaluate -3⁰, we find 3⁰ and then take the opposite. So -3⁰ = -1.

b) Definition of zero exponent

Helpful Hint

Defining a⁰ to be 1 gives a consistent pattern to exponents:

If the exponent is increased by 1 (with base 3) the value of the expression is multiplied by 3.