ObjectivesObjectives

8.1 Laws of Exponents: Multiplying Monomials

Define exponents and powers.Find products of powers.Simplify products of monomials.

Page 370 – Laws of ExponentsMultiplying Monomials

Glossary TermsGlossary Terms

base of a power

coefficient

exponent

monomial

Product-of-Powers Property

8.1 Laws of Exponents: Multiplying Monomials

Base and ExponentBase and Exponent

43

base

exponent

The exponent tells us how many times the base is used as a factor.

Rules and PropertiesRules and Properties

Exponentsxm = x x x . . . x

m factors

For all real numbers x and all positive integers m, when m = 1, xm = x1 = x.

EvaluateEvaluate

28 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 =

256

53 = 5 · 5 · 5 = 125

34 = 3 · 3 · 3 · 3 = 81

61 = 6

Rules and PropertiesRules and Properties

Product-of-Powers Property

For all nonzero real numbers x and all integers m and n,

xm xn = xm+n

8.1 Laws of Exponents: Multiplying Monomials

When you multiply numbers with the same base, add the exponents.

SimplifySimplify

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 Suppose that a colony of bacteria doubles in size every hour. If the colony contains 1000 bacteria at every hour. If the colony contains 1000 bacteria at noon, how many bacteria will the colony contain at noon, how many bacteria will the colony contain at

3 p.m. and 5 p.m. of the same day?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.

Rules and PropertiesRules 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

8.1 Laws of Exponents: Multiplying Monomials

To multiply monomialsTo multiply monomials

1. Remove the parentheses and use the commutative and associative properties to rearrange the terms. Group the constants together, and then group like terms together.

2. Simplify by using the Product-of-Powers Property when appropriate.

SimplifySimplify

(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

Key SkillsKey Skills

Simplify the product of monomials containing exponents.

8.1 Laws of Exponents: Multiplying Monomials

Simplify (5c2d3)(7cd5)

Multiply the constants.

Use the Product-of-Powers Property.

= 35 c2 + 1 d3 + 5

= 35c3d8

= 35 c2 c d3 d5

Simplify

Group terms with the same base.

TOC

The volume of a right rectangular prism can be found by The volume of a right rectangular prism can be found by using the formula V = lwh. Suppose a prism has a length using the formula V = lwh. Suppose a prism has a length

of 2xy, a width of 3xy, and a height of 6xyz. of 2xy, a width of 3xy, and a height of 6xyz. Find the Find the volume.volume.

V = lwh

= (2xy)(3xy)(6xyz)

= 2 · 3 · 6 · x · x · x · y · y · y · z

= 36x³y³z

AssignmentAssignment

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