7-1 multiplying monomials · multiplying monomials is often used when comparing a characteristic of...
TRANSCRIPT
Notes 7-1
Multiplying Monomials
A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. Monomials that are real numbers are called constants.
I. What is a monomial?
Discuss.
A. Identifying monomials
Expression Monomial? Reason
-5 Yes -5 is a real number and an example of a constant
p + q No The expression involves the addition, not the product, of two variables
x Yes Single variables are monomials
B. Examples: Identifying monomials
a. mn2
b. 3x2 + 5x + 7
c. 0.05ab
d. -19x +5
e. -19x
Yes
Yes
Yes
No
No
Today, you will learn three new properties that will help you multiply monomials.
Multiplying monomials is often used when comparing a characteristic of several items, such as acidity of different fruits.
It is also used when determining the probability of something, like guessing the correct answer on a test or winning the lottery (Chapter 12).
Products of powers with the same base can be found by writing each power as a repeated multiplication.
am an = (a a … a) (a a … a)
m factors n factors
= a a … a = am+n
m + n factors
II. Products of Powers
KEY CONCEPT Product of Powers
Words: To multiply two powers that have the same base, add their exponents.
Symbols: For any number a and all integers m and n, am • an = am + n
Example: a4 • a12 = a4 + 12 or a16
A number or variable written without an exponent actually has an exponent of 1.
Remember!
10 = 101
y = y1
Simplify.
A.
Since the powers have the
same base, keep the base
and add the exponents.
a.
Simplify.
Since the powers have
the same base, keep
the base and add the
exponents.
Your turn!
Simplify.
B.
Group powers with the
same base together.
Add the exponents of
powers with the same
base.
b. a2b6a4b9
Your turn!
a2a4b6b9
a2 + 4b6 + 9
a6b15
Group powers with the same base
together.
Add the exponents of powers with
the same base.
Multiply.
C. (6y3)(3y5)
(6y3)(3y5)
18y8
Group factors with like bases
together.
D. (3mn2) (9m2n)
(3mn2)(9m2n)
27m3n3
Multiply.
Group factors with like bases
together.
Multiply.
(6 3)(y3 y5)
(3 9)(m m2)(n2 n)
Multiply. c. (3x3)(6x2)
(3x3)(6x2)
(3 6)(x3 x2)
18x5
Group factors with like bases
together.
Multiply.
Group factors with like bases
together.
Multiply.
d. (2r2t)(5t3)
(2r2t)(5t3)
(2 5)(r2)(t3 t)
10r2t4
Your turn!
When multiplying powers with the same base, keep the base and add the exponents.
x2 x3 = x2+3 = x5
Again…
To find a power of a power, you can use the meaning of exponents.
n factors
= a a … a a a … a … a a … a = amn
= am am … am
m factors m factors m factors
n groups of m factors
III. Power of a Power
KEY CONCEPT Power of a Power
Words: To find the power of a power, multiply the exponents.
Symbols: For any number a and all integers m and n, (am)n = am • n
Example: (k5)9 = k5 • 9 or k45
Simplify.
Use the Power of a Power Property.
Simplify.
Simplify.
Use the Power of a Power Property.
Simplify.
Your turn!
Simplify
B. [(32)3]2
(36)2
312
Power of a Power
Power of a Power
Simplify
Simplify
(32•3)2
36•2
Simplify
B. [(22)2]4
(24)4
216
Power of a Power
Power of a Power
Simplify
Simplify
(22•2)4
24•4
You Try!
Powers of products can be found by using the meaning of an exponent.
(ab)n = ab ab … ab
n factors
= a a … a b b … b = anbn
n factors n factors
KEY CONCEPT Power of a Product
Words: To find the power of a product, find the power of each factor and multiply.
Symbols: For all numbers a and b and any integer m, (ab)m = ambm
Example: (-2xy)3 = (-2)3x3y3 or -8x3y3
Simplify.
Use the Power of a Product Property.
Simplify.
Use the Power of a Product Property.
Simplify.
A.
B.
Caution!
In Example 4A, the negative sign is not part of the base. –(2y)2 = –1(2y)2
Classwork
Workbook
Section 7-1 Page 87