Properties of Exponents
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.
The word exponent comes from the Latin "expo" meaning "out of" and "ponere" meaning "place." While there is some debate, it seems that the Babylonians living in present day Iraq were the first to do work with exponents (dating back to the 23rd century BC or earlier!)
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.
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:
Quotient Property of Exponents
Rather than multiplying, we will now try to divide with exponents.
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:
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:
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:
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.
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.
Power of a Product Rule
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.
Power of a Product Property of Exponents:
It is important to be careful to only use the power of a product property with multiplication inside parentheses. This property does NOT work if there is addition or subtraction. (a+b)m ≠ am + bm These are NOT equal. Beware of this error!
Power of a Quotient Property of Exponents
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.
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:
These five properties are often mixed up in the same problem. Often there is a bit of flexibility as to which property is used first. However, the order of operations still applies to a problem. For this reason, we suggest simplifying inside any parentheses first, then simplify any exponents (using power properties). Finally, simplify any multiplication or division (using product and quotient properties).
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html