# Zero Exponents and Negative Exponents

Negative exponents and zero exponents often show up when applying formulas or simplifying expressions.

In this section, we will define the Negative Exponent Rule and the Zero Exponent Rule and look at a couple of examples.

Negative Exponent Rule: $b^{-2} = \frac{1}{b^n}$

In other words, when there is a negative exponent, we need to create a fraction and put the exponential expression in the denominator and make the exponent positive. For example,

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

But working with negative exponents is just rule of exponents that we need to be able to use when working with exponential expressions.

Example:

Simplify: 3-2

Solution:

3-2 = $3{-2} = \frac{1}{3^2} = \frac{1}{9}$

Example:

Simplify: $\frac{6^{-2}}{7^{-5}}$

Solution:

Apply the Negative Exponent Rule to both the numerator and the denominator.

$\frac{6^{-2}}{7^{-5}} = 6^{-2} \cdot \frac{1}{7^{-5}} = \frac{1}{6^2} \cdot 7^5 = \frac{7^5}{6^2} = \frac{16,807}{36}$

Example:

Simplify: 3-1 + 5-1

Solution:

Apply the Negative Exponent Rule to each term and then add fractions by finding common denominators.

$3^{-1} + 5^{-1} = \frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15}$

Zero Exponent Rule: a0 = 1, a not equal to 0. The expression 00 is indeterminate, or undefined.

In the following example, when we apply the product rule for exponents, we end up with an exponent of zero.

x5x-5 = x5 + (-5) = x0

To help understand the purpose of the zero exponent, we will also rewrite x5x-5 using the negative exponent rule.

x5x-5 = $x^5x^{-5} = \frac{x^5}{x^5} = 1$

The zero exponent indicates that there are no factors of a number.

Example:

Simplify each of the following expressions using the zero exponent rule for exponents. Write each expression using only positive exponents.

a) 30

b) -30 + n0

Solution:

a) Apply the Zero Exponent Rule.

30 = 1

b) Apply the Zero Exponent Rule to each term, and then simplify. The zero exponent on the first term applies to the 3 only and not the negative in front of the 3.

-30 + n0 =-(30) + n0 = - 1 + 1 = 0

4x 0 - 5 0 =
3 -2 =
$\frac{2 \times 10^3}{5 \times 10^{-3}}$