Download 6.2 Multiplying Monomials

6.2 Multiplying Monomials
CORD Math
Mrs. Spitz
Fall 2006
Standard/Objectives
• Standard:
• Objectives: After studying this lesson, you
should be able to:
– Multiply monomials, and
– Simplify expressions involving powers of
monomials.
What’s a monomial?
• A monomial is a number, a variable, or a
product of a number and one or more
variables. Monomials that are real
numbers are constants.
• These are monomials:
-9
y
7a
3y3
½abc5
• These are NOT monomials:
m+n
x/y 3 – 4b
1/x2
7y/9z
Notes:
• Recall that an expression of the form xn is a
power. The base is x and the exponent is n. A
table of powers of 2 is shown below:
20
21
22
23
24
25
26
27
28
29
0
2
4
8
16
32
64 128 256 512
210
1024
Notes:
• Notice that each of the following is true:
4 · 16 = 64
8 · 16 = 128
8 · 32 = 256
22 · 2 4 = 2 6
23 · 2 4 = 2 7
23 · 2 5 = 2 8
Look for a pattern in the products shown. If you consider only the
exponents, you will find that 2 + 4 = 6, 3 + 4 = 7, and 3 + 5 = 8
These examples suggest that you can multiply powers that have the
same base by adding exponents.
Product of Powers Property
• For any number a and all integers m
and n,
am · an = am+n
Ex. 1: Find the measure of the area of a rectangle.
A = lw
= x 3 · x4
= x3+4
= x7
x4
x3
Ex. 2: Simplify
(-5x2)(3x3y2)(
2
5
2
2
3
2
(-5x )(3x y )(
xy4)
5
= (-5· 3 ·
2
5
xy4)
)(x2 · x3 · x)(y2 · y4)
= -6x2+3+1y2+4
= -6x6y6
Step 1: Commutative and associative properties
Step 2: Product of Powers Property
Step 3: Simplify
Notes: Take a look at the examples below:
(52)4 = (52)(52)(52)(52)
= 52+2+2+2
= 58
(x6)2 = (x6)(x6)
= x6+6
= x12
Since (52)4 = 58
and (x6)2 = x12,
these
examples
suggest that
you can find
the power of a
power by
multiplying
exponents.
Power of a power
• For any number a and all integers m
and n,
(am)n = amn
Here are a few more examples
(xy)3 = (xy)(xy)(xy)
= (x · x · x)(y · y · y)
= x3 y 3
(4ab)4 = (4ab) (4ab) (4ab) (4ab)
= (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b)
= 44 a 4 b 4
= 256a4b4
These examples suggest that the power of a product is
the product of the powers.
Power of a Product
• For any number a and all integers m,
(ab)m = ambm
Ex. 3—Find the measure of the volume of
the cube.
V = s3
= (x2y4)3
= (x2)3 · (y4)3
= x 2·3y4·3
= x6y12
x2y4
x2y4
x2y4
Power of a Monomial
• For any number a and b, and any
integers m, n, and p,
(ambn)p = ampbnp
Ex. 4: Simplify (9b4y)2[(-b)2]3
(9b4y)2[(-b)2]3 = 92(b4)2y2(b2)3
= 81b8y2b6
= 81b14y2
Some calculators have a power key labeled yx . You
can use it to find the powers of numbers more
easily. See the next slide.
Ex. 5: Evaluate (0.14)3
Enter: 0.14
yx 3
=
Display will read: 0.002744, so (0.14)3
is about 0.003