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Exponential Properties Involving Products

Add exponents to multiply exponents by other exponents

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Product Rules for Exponents

Suppose you have the expression:

How could you write this expression in a more concise way? 

Product Rules for Exponents

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.

  1. To multiply two terms with the same base, add the exponents.
  1. To raise a product to a power, raise each of the factors to the power.

Let's simplify the following expressions:

The answer can be taken one step further. The base is numerical so the term can be evaluated.

  1. .

  1. .

  1. .

Examples

Example 1

Earlier, you were asked to write the following expression in a more concise way;

 

The expression can be rewritten as . Then, you can use the rules of exponents to simplify the expression to . This is certainly much quicker to write!

Example 2

Simplify the following expression:

 . Here are the steps:

Example 3

Simplify the following expression: 

. Here are the steps:

Example 4

Simplify the following expression:

Here are the steps:

OR

Review

Simplify each of the following expressions, if possible.

Expand and then simplify each of the following expressions.

  1. Hint: Look for a pattern in the previous two problems.

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.1. 

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Vocabulary

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression 32^4, 32 is the base, and 4 is the exponent.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Power

The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".

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