Chapter R- Review
R.2 Integer Exponents
Definition
(i) for any real number a and a positive integer n (1,2,3,4….) a n = a ⋅ a24a
1 ⋅ K3
4 ⋅ n times
3
Example: (2) = 2⋅2⋅2= 8
(ii) for any nonzero real number a: a0 = 1
Example: (-312)0= 1
1
an
Remark: a in the above definition is called the base of the exponent and n is called an exponent or a power
Example: ( −3) −2 = 1 2 = 1
( −3)
9
(iii) for any nonzero real number a and a positive integer n: a −n =
Caution: -32 = -9 but (-3)2 = 9
a n ⋅ a m = a n+m an 1
= a n− m = m− n m a a n m n⋅m (a ) = a
Note that these properties can also be applied in reverse
x 5 = x 2+3 = x 2 ⋅ x 3
( )
x 9 = x 3⋅3 = x 3
3
81x 4 = 34 ⋅ x 4 = (3x) 4
Additional properties of exponents:
−n
b
a
=
a
b m −n a b
= n
−m
b a n
−2
2
32 9
3
2
= = 2 =
2
4
2
3 x −3 y 5
=
y −5 x 3
Caution: Identify the base of the exponent properly. Example:
1
1
x2
=
=
2 x −2 2 ⋅ ( x) −2 2
−2
−1
Example: Simplify the following expression 4 x ( yz )
23 x 4 y
4 x −2 ( yz ) −1
4
41
1
/
= 6 2
= 2 4
= 2 2+4
3 4
1
2 x y x 8 x y ( yz )
8 x yyz 2 x y z
/
R.8 n-th Radicals; Rational Exponents
A square root of a nonnegative number a is a number b such that b2 = a.
Example : square roots of 25 are 5 and -5 since 52 = 25 and (-5)2 = 25
Square root of 0 is 0, since 02 = 0
Square root of – 4 does not exist (in the real number system), since there is no real number that squared gives (-4)
The principal square root, or radical number a is called a radicand.
25 = 5 ,
Example
0 = 0,
, of a nonnegative number a is a nonnegative number b such that b2 = a. The
− 4 not defined
Remark : a) the principal square root of a is often called the square root of a or radical of a
b) the square root of a is NEVER negative
Properties of radicals
( a)
2
( 3)
2
= a,
a≥0
( −5) 2 =| −5 |= 5
a 2 =| a | a ⋅ b = a ⋅ b, a =3
a
=
b
b
am =
( a) ,
,
9⋅5 = 9 ⋅ 5 = 3 5
a, b ≥ 0
4
=
25
m
a≥0
25
16 3 =
b≠0
4
( 16 )
=
3
2
5
= 4 3 = 64
To simplify a radical means to remove all factors that are perfect squares
Example : 12 x 5 = 3 ⋅ 4 ⋅ x 4 ⋅ x =
Note that
x8 =
(x )
4 2
( )
4 ⋅ x2
2
⋅ 3x = 4
= x 4 and, in general,
x 5 = x 4 ⋅ x = x 2 x and, in general,
(x )
2 2
⋅ 3x = 2 x 2 3x
x even = x even / 2 x odd = x ( odd −1) / 2 x
If two expressions contain the same radical (same index and same radicand), then that radical can be factored out and the two expressions combined.
Example: 3 12 − 4 27 = 3 4 ⋅ 3 − 4 9 ⋅ 3 = 3 ⋅ 2 3 − 4 ⋅ 3 3 = 6 3 − 12 3 = 3 (6 − 12) = −6 3
To rationalize the denominator (or numerator) is to eliminate the radical from the denominator (or numerator) through some algebraic operations
Teaching Notes TOYS “R” US, AS OF OCTOBER, 2004 Case Uses & Objectives The Toys “R” Us case can be used as an introductory case to accompany discussion of Chapter 1, as an overview of the many decisions and actions an organization has to undertake to sustain a competitive advantage. This case can be also used to augment discussions of strategic analysis, specifically both internal and external environmental analysis (Chapters 2 & 3 in Dess, Lumpkin & Eisner); and strategic formulation, at both…
now: Read and turn in quizzes for Chapters 7&8 Complete and submit Homework #3 9/22/14 Agenda for the session: Guest speaker: Beth Harrelson Break Content Review: Chapter 7 and 8 Review: Homework #3 Something new Preview of Session 5 (2/20/13) Guest Speaker Beth Harrelson, CCIM Q 10 Professional Mortgage of North Carolina, LLC Raleigh, NC Let’s take a 10-minute break Discussion Board – please post Get some fresh air! Appraisal: Part One Chapter 7 key concepts Real estate value:…
INTB 311 Diversity & Intercultural Communication Chapter 1: Culture B AC K S TAG E , O N S TAG E , C U LT U R E S H O C K , R E V E R S E C U LT U R E S H O C K , G E N E RA L I Z AT I O N S , S T E R E O T Y P E S , BIAS, PREJUDICE, ETHNOCENTRISM, ASSUMPTION OF SUPERIORITY, ASSUMPTION OF UNIVERSALITY, SELFR E F E R E N C E C R I T E R I A , C U LT U RA L I N T E L L I G E N C E . INTB 311 Diversity & Intercultural Communication Culture & Communication Why is understanding culture important…
CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING Systems, Society and Sustainability The global challenge of sustainable development requires solutions and mindsets that bridge traditional divisions between nature and culture, and the technical and social sciences. Sustainable development requires that engineers and other professionals are able to include social and ecological considerations alongside technical and economic requirements in managing projects and infrastructure. This course outlines…
110 – 111 “Sometimes I could cope….” to end of chapter. c) pp 117 – 118 “Devil…again be virtuous” d) pp 131 – 132 “I lay on my straw….more for themselves”. e) Pp 137 – 138 “My thoughts now become…” to end of chapter. f) Pp. 142 – 144 “ My days were spent……whom all men disowned.” g) Pp 155 – 158 “As I read….but am solitary and abhorred.” h) Pp 174 – 175 “As I fixed my eyes….this being you must create.” i) Pp 179 – 180 “ “How inconstant are you r feelings…from which I am now excluded.” j) Pp…