Table of Properties
Properties of Addition and Multiplication
Inverse Properties (pp. 30, 247)
The sum of a number and its additive inverse, or opposite, is 0.
The product of a nonzero number and its multiplicative
inverse, or reciprocal, is 1.
a (a) 0
a b
p 1 (a 0, b 0)
b a
Commutative Properties (p. 63)
In a sum, you can add the numbers in any order.
In a product, you can multiply the numbers in any order.
abba
ab ba
Associative Properties (p. 63)
Changing the grouping of the numbers in a sum does not
change the sum.
Changing the grouping of the numbers in a product does
not change the product.
(a b) c a (b c)
(ab)c a(bc)
Identity Properties (p. 64)
The sum of a number and the additive identity, 0, is the number.
The product of a number and the multiplicative identity, 1,
is the number.
a0a
ap1a
Distributive Property (p. 71)
You can multiply a number and a sum by multiplying
each term of the sum by the number and then adding
these products. The same property applies to subtraction.
a(b c) ab ac
a(b c) ab ac
Table of Properties
Properties of Equality
820
Subtracting the same number from each side of an equation
produces an equivalent equation.
If x a b, then
x a a b a,
or x b a.
Addition Property of Equality (p. 92)
Adding the same number to each side of an equation
produces an equivalent equation.
If x a b, then
x a a b a,
or x b a.
Division Property of Equality (p. 97)
Dividing each side of an equation by the same
nonzero number produces an equivalent equation.
If ax b and a 0,
then a a, or x a.
Multiplication Property of Equality (p. 98)
If b and a 0, then
Multiplying each side of an equation by the same
nonzero number produces an equivalent equation.
a p a a p b, or x ab.
Subtraction Property of Equality (p. 91)
Table of Properties
ax
x
a
x
b
b
Properties of Inequality
If a < b, then a c < b c
and a c < b c.
Addition and Subtraction Properties of Inequality (p. 139)
Adding or subtracting the same number on each
side of an inequality produces an equivalent inequality.
If a > b, then a c > b c
and a c > b c.
Multiplication and Division Properties of Inequality (pp. 144, 145)
If a < b and c > 0,
Multiplying or dividing each side of an inequality by a positive
number produces an equivalent inequality. Multiplying or
dividing each side of an inequality by a negative number
and reversing the direction of the inequality symbol produces
an equivalent inequality.
then ac < bc and a < b.
c
c
If a < b and c < 0,
then ac > bc and a > b.
c
c
Properties of Exponents
Product of Powers Property (p. 194)
To multiply powers with the same base, add their exponents.
am p an am n
Quotient of Powers Property (p. 195)
am
am n, a 0
an
To divide powers with the same nonzero base, subtract the
denominator’s exponent from the numerator’s exponent.
Power of a Product Property (p. 674)
To find the power of a product, find the power of each factor
and multiply.
(ab)m ambm
Power of a Quotient Property (p. 674)
冢冣
To find the power of a quotient, find the power of the numerator and
the power of the denominator and divide.
Power of a Power Property (p. 675)
a
b
m
am
b
, b 0
m
Other Properties
Cross Products Property (p. 281)
The cross products of a proportion are equal.
If a c (b, d 0), then ad bc.
b
d
Table of Properties
(am)n amn
To find the power of a power, multiply the exponents.
Product Property of Square Roots (p. 458)
The square root of a product is equal to the product of
the square roots of the factors.
兹ab
苶 兹a
苶 p 兹b
苶, a ≥ 0 and b ≥ 0
Quotient Properties of Square Roots (p. 459)
兹a
苶
, a ≥ 0 and b > 0
冪莦ab 兹b
苶
The square root of a quotient is equal to the quotient
of the square root of the numerator and the square root
of the denominator.
Table of Properties
821