6.1: Product Rules for Exponents
Suppose you have the expression:
How could you write this expression in a more concise way?
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Guidance
In the expression , the is called the base and the is called the exponent. Exponents are often referred to as powers. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:
- .
There are many rules that have to do with exponents (often called the Laws of Exponents) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn two rules that have to do with exponents and products.
RULE: To multiply two terms with the same base, add the exponents.
RULE: To raise a product to a power, raise each of the factors to the power.
Example A
Evaluate .
Solution:
The answer can be taken one step further. The base is numerical so the term can be evaluated.
Example B
Evaluate .
Solution:
Example C
Evaluate .
Solution:
Example D
Evaluate .
Solution:
Concept Problem Revisited
can be rewritten as . Then, you can use the rules of exponents to simplify the expression to . This is certainly much quicker to write!
Vocabulary
- Base
- In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression , ‘2’ is the base. In the expression , ‘’ is the base.
- Exponent
- In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
- In the expression , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here:
- In the expression , ‘4’ is the exponent. It means to multiply times itself 4 times as shown here: .
- Laws of Exponents
- The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.
Guided Practice
Evaluate each of the following:
1.
2.
3.
4.
Answers:
1.
2.
3.
- The answer can be taken one step further. The base of each factor is numerical so each term can be evaluated. The final answer will be the product of the two answers.
4.
Practice
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Here you'll learn how to multiply two terms with the same base and how to find the power of a product.