CHAPTER 7 Powers, Roots, and Scientific Notation
7.1 Use Multiplication Properties of Exponents
Learning Objectives
By the end of this section, you will be able to:
- Simplify expressions with exponents
- Simplify expressions using the Product Property for Exponents
- Simplify expressions using the Power Property for Exponents
- Simplify expressions using the Product to a Power Property
- Simplify expressions by applying several properties
- Multiply monomials
Simplify Expressions with Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, means to multiply 2 by itself 4 times, so means 2 · 2 · 2 · 2
Let’s review the vocabulary for expressions with exponents.
Exponential Notation
This is read to the power.
In the expression , the exponent tells us how many times we use the base as a factor.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.
EXAMPLE 1
Simplify: a) b) c) d) .
| a) | |
| Multiply three factors of 4. | 4 · 4 · 4 |
| Simplify. | |
| b) | |
| Multiply one factor of 7. | |
| c) | |
| Multiply two factors. | |
| Simplify. | |
| d) | |
| Multiply two factors. | |
| Simplify. |
TRY IT 1.1
Simplify: a) b) c) d) .
Show answer
a) 216 b) c) d) 0.1849
TRY IT 1.2
Simplify: a) b) c) d) .
Show answer
a) b) 21 c) d)
EXAMPLE 2
Simplify: a) b) .
| a) | |
| Multiply four factors of . | |
| Simplify. | |
| b) | |
| Multiply four factors of 5. | -(5 · 5 · 5 · 5) |
| Simplify. |
TRY IT 2.1
Simplify: a) b) .
Show answer
a) b)
TRY IT 2.2
Simplify: a) b) .
Show answer
a) b)
Notice the similarities and differences in (Example 2) a) and (Example 2) b)! Why are the answers different? As we follow the order of operations in part a) the parentheses tell us to raise the to the 4th power. In part b) we raise just the 5 to the 4th power and then take the opposite.
Simplify Expressions Using the Product Property for Exponents
You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
We’ll derive the properties of exponents by looking for patterns in several examples.
First, we will look at an example that leads to the Product Property.
| What does this mean? How many factors altogether? |
|
| So, we have | |
| Notice that 5 is the sum of the exponents, 2 and 3. |
We write:
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
Product Property for Exponents
If is a real number, and and are counting numbers, then
To multiply with like bases, add the exponents.
An example with numbers helps to verify this property.
EXAMPLE 3
Simplify: .
| Use the product property, am \cdot an = am+n. | |
| Simplify. |
TRY IT 3.1
Simplify: .
Show answer
TRY IT 3.2
Simplify: .
Show answer
EXAMPLE 4
Simplify: a) b) .
-
Use the product