In Exponent Basics we worked with whole number exponents. The whole numbers are the set of numbers {0, 1, 2, 3, 4, ...}. The whole numbers are the positive integers, plus zero.

On this page, we will be examing exponents that are negative integers.

 Negative Integer Exponents

The use of a negative integer exponent has special meaning and a rule explaining its use.

 For any non-zero number x, and for any positive integer n,

Let's take a closer look at why this Rule is true:

One of the Laws of Exponents is that xm • xn = xm+n.
"When multiplying exponential expressions, if the bases are the same, add the exponents."

If we apply this law to work with a negative exponent, we get 43 • 4-3 = 43+(-3) = 40 = 1.
This application shows us that 43 • 4-3 = 1, which means that 4-3 must the multiplicative identity of 43. Therefore, 4-3 must be a fraction and it must be the reciprocal of 43.
Consequently,
.

There are three important concepts at work in this Rule:

For any non-zero number x, and for any positive integer, n:

Remember, any number (or expression) with a negative exponent ends up
on the opposite side of the fraction bar as a positive exponent.

The use of a positive exponent is an application of repeated multiplication by the base:
43 = 4 • 4 • 4 = 64.
The use of a negative exponent produces the opposite of repeated multiplication.
It can be thought of as a form of repeated division by the base:
4-3 = 1 ÷ 4 ÷ 4 ÷ 4 = 0.015625

Examples:

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