Properties
Properties of Real Numbers
Let a, b, and c be real numbers.
Addition
Multiplication
Closure Property (p. 3)
a 1 b is a real number.
ab is a real number.
Commutative Property (p. 3)
a1b5b1a
ab 5 ba
Associative Property (p. 3)
(a 1 b) 1 c 5 a 1 (b 1 c)
(ab)c 5 a(bc)
Identity Property (p. 3)
a 1 0 5 a, 0 1 a 5 a
a p 1 5 a, 1 p a 5 a
Inverse Property (p. 3)
a 1 (2a) 5 0
1
ap}
a 5 1, a Þ 0
Distributive Property (p. 3)
The distributive property involves both addition and multiplication:
a(b 1 c) 5 ab 1 ac
Zero Product Property (p. 253)
Let A and B be real numbers or algebraic expressions.
If AB 5 0, then A 5 0 or B 5 0.
Properties of Matrices
Let A, B, and C be matrices, and let k be a scalar.
(A 1 B) 1 C 5 A 1 (B 1 C)
Commutative Property of Addition (p. 188)
A1B5B1A
Distributive Property of Addition (p. 188)
k(A 1 B) 5 kA 1 kB
Distributive Property of Subtraction (p. 188)
k(A 2 B) 5 kA 2 kB
Associative Property of Matrix Multiplication (p. 197)
(AB)C 5 A(BC)
Left Distributive Property of Matrix Multiplication (p. 197)
A(B 1 C) 5 AB 1 AC
Right Distributive Property of Matrix Multiplication (p. 197)
(A 1 B)C 5 AC 1 BC
Associative Property of Scalar Multiplication (p. 197)
TABLES
Associative Property of Addition (p. 188)
k(AB) 5 (kA)B 5 A(kB)
Multiplicative Identity (p. 210)
An n 3 n matrix with 1’s on the main diagonal
and 0’s elsewhere is an identity matrix, denoted I.
For any n 3 n matrix A, AI 5 IA 5 A.
Inverse Matrices (p. 210)
If the determinant of an n 3 n matrix A is
nonzero, then A has an inverse, denoted A21, such
that AA21 5 A21 A 5 I.
Properties of Exponents
Let a and b be real numbers, and let m and n be integers.
Product of Powers Property (p. 330)
am p an 5 am 1 n
Power of a Power Property (p. 330)
(am ) n 5 amn
Power of a Product Property (p. 330)
(ab) m 5 ambm
Negative Exponent Property (p. 330)
1 ,aÞ0
a2m 5 }
m
a
0
Zero Exponent Property (p. 330)
a 5 1, a Þ 0
Quotient of Powers Property (p. 330)
}
n 5a
Power of a Quotient Property (p. 330)
1 }b 2
am
a
a m
m2n
,aÞ0
m
a
5}
m, b Þ 0
b
Tables
n2pe-9040.indd 1033
1033
10/14/05 11:06:11 AM
Properties of Radicals and Rational Exponents
Number of Real nth Roots
Let n be an integer greater than 1, and let a be a real number.
(p. 414)
Radicals and Rational
Exponents (p. 415)
•
•
•
•
n}
If n is odd, then a has one real nth root: Ï a 5 a1/n
n}
If n is even and a > 0, then a has two real nth roots: 6 Ï a 5 6a1/n
}
n
If n is even and a 5 0, then a has one nth root: Ï0 5 01/n 5 0
If n is even and a < 0, then a has no real nth roots.
Let a1/n be an nth root of a, and let m be a positive integer.
n}
• am/n 5 (a1/n ) m 5 1 Ï a 2m
1
1
1
• a2m/n 5 }
5}
5}
,aÞ0
n} m
m/n
1 a1/n 2m
a
1 Ïa 2
Properties of Rational
Exponents (p. 420)
All of the properties of exponents listed on the previous page apply to rational
exponents as well as integer exponents.
Product and Quotient
Properties of Radicals (p. 421)
Let n be an integer greater than 1, and let a and b be positive real
n}
n}
n}
numbers. Then Ïa p b 5 Ï a p Ïb and
n
}
n}
Ïa
a
}5}
n }.
Ïb
Ïb
Properties of Logarithms
TABLES
Let a, b, c, m, n, x, and y be positive real numbers such that
b Þ 1 and c Þ 1.
Logarithms and Exponents (p. 499)
log b y 5 x if and only if b x 5 y
Special Logarithm Values (p. 499)
log b 1 5 0 because b 0 5 1 and log b b 5 1 because b1 5 b
Common and Natural Logarithms (p. 500)
log10 x 5 log x and loge x 5 ln x
Product Property of Logarithms (p. 507)
log b mn 5 log b m 1 log b n
Quotient Property of Logarithms (p. 507)
m
log b }
n 5 log b m 2 log b n
Power Property of Logarithms (p. 507)
log b mn 5 n log b m
Change of Base (p. 508)
logc a 5 }
logb a
logb c
Properties of Functions
Operations on Functions
(pp. 428, 430)
Let f and g be any two functions. A new function h can be defined using
any of the following operations.
Addition:
Subtraction:
Multiplication:
h(x) 5 f(x) 1 g(x)
h(x) 5 f(x) 2 g(x)
h(x) 5 f(x) p g(x)
Division:
h(x) 5 }
Composition:
h(x) 5 g(f(x))
f(x)
g(x)
For addition, subtraction, multiplication, and division, the domain
of h consists of the x-values that are in the domains of both f and g.
Additionally, the domain of the quotient does not include x-values for
which g(x) 5 0.
For composition, the domain of h is the set of all x-values such that x is in
the domain of f and f(x) is in the domain of g.
Inverse Functions (p. 438)
Functions f and g are inverses of each other provided:
f(g(x)) 5 x and g(f(x)) 5 x
1034 Student Resources
n2pe-9040.indd 1034
10/14/05 11:06:12 AM