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Common Core Standard    N.RN.2

# Negative exponents rules and examples

​To understand negative exponents think positive. When a number contains an exponent it tells you the number of times that the number will be multiplied by itself.   For example 2^3 = 2x2x2 = 8

What do you do when this exponent is negative? For example, 2^-3 becomes 1/2^3.

The negative exponents means how many times to divide by this number.

2^-3 = 1÷2÷2÷2= .125

5^2 = 1÷5÷5 = .04

An easier method is to take the number to the power of the positive exponent, and then take the reciprocal of this number.

4^-2 = Take 4 to the power of 2  4*4 =16 then take the reciprocal = 1/16 = .o625

5^-3= Take 5 to the power of 3, 125 then take the reciprocal = 1/125 = .oo8

You can think of a negative exponent as the positive reciprocal of the number taken to the power of the exponent.

Negative exponent rule: A negative exponent located in the numerator is changed to positive, and is moved to the denominator. Conversely, a negative exponent located in the denominator is changed to positive, and moved to the numerator.

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## Negative Exponents practice

In these examples the problems are rational exponents, but in the process of simplying the fractional exponent you end up with a negative exponent..

## Simplifying Expressions with Negative Exponents

• How do you simplify with negative exponents?
• When simplifying a rational exponent how do you treat the negative exponent?
• How do you simplify 2x^-2?
• What do the parenthesis mean with (2x)^-?
• What is the rule of wrong position?
• How do you simplify a fraction with exponents in the denominator?
• How do you simplify a fraction raised to a negative power?

Example 2 is answered at 1:17 in the video above
This problem is answered at 1:05 in the video above
What about these? Remember the exponent only applies to the variable.

## Negative exponents and parentheses

Whenever you have an exponent inside parenthesis. just apply the exponent to everything inside them.
Because there are not parenthesis, the exponent only applies to the variable.
The 3 remains in the numerator because the exponent only applies to the variable.
= .111
= .015625
= .03125
= .01234

## Negative exponents and fractions

The negative exponent rules also work with fractions.

Example 1
The negative exponent in the numerator is changed to a positive exponent, and moved to the denominator,and the negative exponent in the denominator is changed to a positive exponent and moved to the numerator.
Example 2 Negative exponents and fractions
Example 3 Fractions with negative exponents
​Remember the exponent only applies to the variable,unless parenthesis are used.

## Negative exponents and scientific notation

Negative exponents are also used with scientific notation.
Scientific notation is a method used to write really large, or very small numbers more efficiently. For instance,

105000000 can be written using scientific notation  as 1.05 x 10^8

3.05 x10^9 becomes 3050000000  This would represent a fairly large number.

Negative exponents allow you to write really small numbers.

1.05 x 10^-8 = 0.0000000105

0.000000000000354  can be written using scientific notation as 3.54 x 10^-13 using scientific notation