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Applying the Properties of Exponents

Author: Sophia
what's covered
  1. Properties of Exponents
  2. Cautions when Applying the Properties
  3. Applying the Properties of Exponents

1. Properties of Exponents

There are several properties of exponents involving multiplication, division, and powers that can help us simplify expressions containing exponents. They are:

The Product Power: a to the power of n • a to the power of m equals a to the power of n plus m end exponent

The Quotient Power: a to the power of n over a to the power of m equals a to the power of n minus m end exponent

The Power of Powers: left parenthesis a to the power of n right parenthesis to the power of m equals a to the power of n m end exponent

Power of a Product: left parenthesis a b right parenthesis to the power of n equals a to the power of n • b to the power of n

Power of a Quotient: left parenthesis a over b right parenthesis to the power of n equals a to the power of n over b to the power of n

Negative Exponents: a to the power of negative n end exponent equals 1 over a to the power of n


2. Cautions with Applying these Properties

There are a couple of things to watch out for when applying these properties. First, it is important to remember that the product and quotient powers can only be applied with the bases are the same. We cannot simply add the exponents if the bases are not the same.

hint
a to the power of n • b to the power of m not equal to left parenthesis a b right parenthesis to the power of n plus m end exponent
Secondly, the product and quotient of power properties only work with multiplication and division; they do not work with addition or subtraction.
hint
open parentheses a space plus space b close parentheses to the power of n space not identical to space a to the power of n space plus space b to the power of n


3. Applying the Properties of Exponents

Let's go through some examples where we can use different properties of exponents to simplify expressions:

EXAMPLE

File:7535-e1.png
If we see factors that appear in both the numerator and denominator, we cancel them out. This means that we reduce the power in both the numerator and denominator. If we cancel all factors in the denominator, the denominator is 1, and doesn't need to be written.

EXAMPLE

File:7536-e2.png

summary
Use the properties of exponents to simplify expressions and solve equations. However, consider the following cautions when applying the properties. You can only add or subtract the exponents in multiplication or division if the bases are the same. And you can only distribute an exponent across factors and not terms.

Formulas to Know
Power of a Power Property of Exponents

open parentheses a to the power of n close parentheses to the power of m equals a to the power of n m end exponent

Power of a Product Property of Exponents

open parentheses a b close parentheses to the power of n equals a to the power of n times b to the power of n

Power of a Quotient Property of Exponents

open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n

Product Property of Exponents

a to the power of n times a to the power of m equals a to the power of n plus m end exponent

Property of Negative Exponents

a to the power of negative n end exponent equals 1 over a to the power of n

Quotient Property of Exponents

a to the power of n over a to the power of m equals a to the power of n minus m end exponent