Before You Start
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 the important properties.
[DID YOU KNOW ICON]
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 Iraq were the first to do work with exponents (dating back to the 23rd century BC or earlier!)
Product of Powers Property
Let's consider multiplication with exponents when the bases are the same:
A quicker method to arrive at our answer would have been to just add the exponents. This is known as the product of powers property.
The product of powers property can be used to simplify many problems. If exponential expressions with the same base are multiplied, we can add the exponents. Here is an example:
Quotient of Powers Property 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 of powers property:
The quotient of powers property can similarly be used to simplify exponent problems by subtracting exponents on like-variables. Here is an example:
Power of Power Property
A third property we will look at will have an exponent problem raised to a second 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.
This property is often combined with two other properties: power of a product, and power of a quotient.
Power of a Product Property
A quicker method to arrive at the solution would have been to take the exponent of three and put it on each factor in parenthesis. This is known as the power of a product property.
It is important to be careful to only use the power of a product rule with multiplication inside parenthesis. This property does NOT work if there is addition or subtraction.
These are NOT equal. Beware of this error!
Power of a Quotient Property
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.
The power of a power, product and quotient rules 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, order of operations still applies to a problem. For this reason, we suggest simplifying inside any parentheses first, then simplify any exponents (using power rules). Finally, simplify any multiplication or division (using product and quotient rules).