EDGE Math 8

Zero and Negative Integer Exponents

The laws of exponents previously discussed are limited to positive integer exponents. The same laws will now be extended to zero and negative integer exponents.

Zero Exponents
To define $a^{0},$ where $a$ is a nonzero real number, consider the Product Rule. If the Product Rule must hold when $n = 0$ , you will have

a^{m} \cdot a^{0} = a^{m + 0} = a^{m} .

Hence, $a^{m} \cdot a^{0} = a^{m} .$ For this equation to be true, $a^{0}$ should be equal to 1. This leads to the following definition:

Definition

If $a$ is a nonzero real number, then $a^{0} = 1.$

Example

Simplify each expression. Assume that the variables represent nonzero real numbers.

1. $5^{0}$

2. ${(- 10)}^{0}$

3. ${(6 x)}^{0}$

4. $7 x^{0}$

Solution

Use the definition of zero exponent.

1. $5^{0} = 1$

2. ${(- 10)}^{0} = 1$

3. ${(6 x)}^{0} = 6^{0} x^{0} = 1 \cdot 1 = 1$

4. $7 x^{0} = 7 \cdot 1 = 7$

Take note that there is no real number that is equal to $0^{0};$ that is, $0^{0}$ is undefined.