Problems with exponents can often be simplified using a few basic exponent properties. Exponents represent repeated multiplication. We will use this fact to discover important properties.
1a. Product Property of Exponents
A quicker method to arrive at our answer would have been to just add the exponents. This is known as the product property of exponents.
The important thing here is that the expressions must have the same base. If exponential expressions with the same base are multiplied together, we can add the exponents. Here is another example:

Same base, add the exponents  

Our Solution 
1b. Quotient Property of Exponents
Rather than multiplying, we will now try to divide with exponents.

Expand Exponents  

Divide out two of the  

Convert to Exponents  

Our Solution 
A quicker method to arrive at the solution would have been to just subtract the exponents. This is known as the quotient property of exponents:
Just like with the product property, it is important to note that is only holds true when the bases are the same. Here is an example:

Same base, subtract the exponents  

Our Solution 
1c. Power of a Power Property of Exponents
A third property we will look at will have an exponent raised to another exponent. This is investigated in the following example:

This means we have three times  

Add exponents  

Our Solution 
A quicker method to arrive at the solution would have been to just multiply the exponents. This is known as the power of a power property of exponents.
This property is often combined with two other properties: power of a product, and power of a quotient. We will look at these properties next.
1d. Power of a Product Rule

This means we have three times  

Three and three can be written with exponents  

Our Solution 
A quicker method to arrive at the solution would have been to take the exponent of three and put it on each factor in parentheses. This is known as the power of a product property of exponents.
1e. Power of a Quotient Property of Exponents

This means we have the fraction three times  

Multiply fractions across the top and bottom, using exponents  

Our Solution 
A quicker method to arrive at the solution would have been to put the exponent on every factor in both the numerator and denominator. This is known as the power of a quotient property of exponents.
The power of a power, product, and quotient properties of exponents are often used together to simplify expressions. This is shown in the following examples:

Put the exponent of on each factor, multiplying powers  

Our Solution  
 

Put the exponent of on each factor, multiplying powers  

Our Solution 
Product Rule of Exponents  

Quotient Rule of Exponents  
Power of a Power Rule of Exponents  
Power of a Product Rule of Exponents  
Power of a Quotient Rule of Exponents 
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html