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. {..., -5, -4, -3, -2, -1}.

Negative Integer Exponents

An value raised to a negative exponent is equal to the number one divided by the value with the sign of the exponent changed to positive.


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

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There are three important concepts at work in this Rule:

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

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


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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

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1.  neg1  
The negative 1 exponent indicates that the value is the same as 1 over 3 raised to a power of positive 1.   
2.  neg2

3.  neg3
Be sure of keep the negative base in the set of parentheses to avoid calculation errors.
4.  neg4
This example is working with a decimal base. The same process applies.
5. neg5
Working with a fraction as the base can be more complicated. When applying the process for negative exponents, a "complex" fraction is formed (a fraction within a fraction). Remember that the fraction bar means divide, when rewriting the complex fraction.
6.  neg6
This is similar to scientific notation, which would be 4.0 x 10-3.
7.  neg7
Negative exponents can be also used with variables. Just imagine the variable to be a numeric value and apply the process for negative exponents.

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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, 4recip

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For calculator help with exponents
click here.


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