Download 8.1 Laws of Exponents: Multiplying Monomials

Page 370 – Laws of Exponents
Multiplying Monomials
8.1 Laws of Exponents: Multiplying Monomials
Objectives
Define exponents and powers.
 Find products of powers.
 Simplify products of monomials.

8.1 Laws of Exponents: Multiplying Monomials
Glossary Terms
base of a power
coefficient
exponent
monomial
Product-of-Powers Property
Base and Exponent
exponent
4
base
3
The exponent tells us how many times the base
is used as a factor.
Rules and Properties
Exponents
xm = x  x  x  . . .  x
m factors
For all real numbers x and all positive
integers m, when m = 1, xm = x1 = x.
Evaluate
28 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 256
=
3
5 = 5 · 5 · 5 = 125
34 = 3 · 3 · 3 · 3 = 81
61 = 6
8.1 Laws of Exponents: Multiplying Monomials
Rules and Properties
Product-of-Powers Property
For all nonzero real numbers x and all
integers m and n,
xm  xn = xm+n
When you multiply numbers with the same
base, add the exponents.
Simplify
23 · 24 = 2 · 2 · 2 · 2 · 2 · 2 · 2 = 27 = 128
8 · 8³ = 8 · 8 · 8 · 8 = 84 = 4096
y2 · y5 = y2 + 5 = y7
5m · 5p = 5m + p
Suppose that a colony of bacteria doubles in size
every hour. If the colony contains 1000 bacteria at
noon, how many bacteria will the colony contain at
3 p.m. and 5 p.m. of the same day?
Between noon and 3 p.m. there are 3 hours so there will
be 1000 · 2³, or 8000 bacteria. The 2 stands for the doubling
and the 3 stands for 3 hours.
At 5 p.m., 2 hours later, the bacteria will double 2 more times.
There will be (1000 · 2³) · 2², or 1000 · 23 + 2, or 1000 · 25,
32,000 bacteria in the colony.
8.1 Laws of Exponents: Multiplying Monomials
Rules and Properties
Definition of Monomial
monomial: a constant, variable, or a
product of a constant and one or more
variables
Coefficient – the number that goes with
the variable
To multiply monomials
1.
2.
Remove the parentheses and use the
commutative and associative properties to
rearrange the terms. Group the constants
together, and then group like terms together.
Simplify by using the Product-of-Powers
Property when appropriate.
Simplify
(5t)(-30t²) = 5 · (-30) · t · t² = -150t³
(-4a²b)(-ac²)(3b²c²) = -4 · (-1) · 3 · a² ·a · b · b² · c² · c² =
12a³b³c4
(3m²)(60mp²) = 3 · 60 · m² · m · p² = 180m³p²
(8xz)(-10y)(-2yz²) = 8 · (-10) · (-2) · x · y · y ·z · z² = 160xy²z³
8.1 Laws of Exponents: Multiplying Monomials
Key Skills
Simplify the product of monomials
containing exponents. Simplify (5c2d3)(7cd5)
Group terms
Multiply
the with the
same
base.
constants.
= 35  c2  c  d3  d5
Use the Product-ofPowers Property.
= 35  c2 + 1  d3 + 5
Simplify
= 35c3d8
TOC
The volume of a right rectangular prism can be found by
using the formula V = lwh. Suppose a prism has a length
of 2xy, a width of 3xy, and a height of 6xyz. Find the
volume.
V = lwh
= (2xy)(3xy)(6xyz)
=2·3·6·x·x·x·y·y·y·z
= 36x³y³z
Assignment

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