Name __________________________________
Period __________
Date:
4-2 Using
Laws of
Exponents
Topic:
Standard: 8.EE.1
Objective:
Essential Question: You know that multiplication is
repeated addition. For example,
. How is
exponentiation related to multiplication?
Know and apply the properties of integer exponents to generate
2
5
3
3
equivalent numerical expressions. For example, 3 ×3 =3 =1/3 =1/27.
To use the laws of exponents to multiply a polynomial by a
monomial.
In this lesson, you will review the laws of exponents that are
used when multiplying polynomials.
Laws of Exponents
Let a and b be real numbers and m and n be positive
integers. Then:
1.
Summary
2. (
)
3. (
)
First Law: Without the first law you would have to simplify a product of
powers, such as c5·c3 by counting factors:
5 factors
5 factors
(
3 factors
5 factors
)(
)
8 factors
5 factors
By using the first law you can write
Second Law: Using the second law you can simplify the power of a product:
(
)
Third Law: Using the third law you can simplify a power of a power:
(
Proof of Second
Law:
)
We’ll prove the second law by counting factors:
1. (
2.
)
(
(
)(
)(
)
)(
(
)(
)
Definition of a
power
)
Commutative
and associative
properties of
multiplication
3.
You can use the laws of exponents along with the commutative
and associative properties of multiplication to simplify
products and powers like the following examples.
2
Example 1:
Simplify
a. (
)(
)
Use the first law of exponents and the fact that
(
b.
(
)(
)
(
)(
)(
.
)
)
Use the second law of exponents then apply the third.
(
c.
(
)
( )
)
(
Use the fact that
exponents.
(
)
[(
)
(
) (
)
and the second law of
]
)
3
Exercise 1:
Simplify
a. (
b.
(
c.
(
)(
)(
)
)
)
4
Example 2:
Simplify
(
a.
(
b.
Exercise 2:
)
)
(
)
(
)
(
) ( )
Simplify
a. (
b.
(
)(
)
)
5
To multiply a polynomial by a binomial, use the distributive property.
Example 3:
(
Simplify
(
Exercise 3:
)
(
)
(
)
(
)
(
)
Simplify
a.
(
)
(
b.
Example 4:
)
)
Simplify. Assume that variable exponents represent positive integers.
a. (
) (
)
(
) (
)
6
b.
Exercise 4:
(
)
(
)
Simplify. Assume that variable exponents represent positive
integers.
a.
a.
(
)
Class work:
p 172 Oral Exercises: 1-27
Homework:
p 173 Written Exercises: 2-38 even
7