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Chapter 5: Rational Numbers as Fractions
5.3 Multiplication and division of rational numbers
5.3.1. Vocabulary
5.3.1.1. reciprocal – multiplicative inverse of a rational number
5.3.2. Multiplication of rational numbers
a
c
5.3.2.1. Definition of multiplication of rational numbers: If
and
are any rational
b
d
a b ac
numbers, then  
b d bd
5.3.3. Properties of multiplication of rational numbers
5.3.3.1. Properties
5.3.3.1.1. Multiplication identity of rational numbers: The number 1 is the unique number
such that for every rational number ba , 1 ba   ba   1  ba
5.3.3.1.2. Multiplicative inverse of rational numbers: For any non-zero rational number
a b
, a is the unique rational number such that ba  ba  ba  ba  1
b
5.3.3.1.3. The multiplicative inverse of
5.3.3.2.
a
b
is also called the reciprocal of
a
b
Properties
5.3.3.2.1. Distributive property of multiplication over addition for rational numbers: If
a
,
b
e
c
ac e a c a e
, and
are any rational numbers, then           
d
f
b  d f  b d b f 
a c
5.3.3.3. Multiplication property of equality for rational numbers: If , , are any rational
b d
e
a c
a e c e
numbers such that  , and
is any rational number, then   
b d
f
b f d f
a
5.3.3.4. Multiplication property of zero for rational numbers: If
is any rational number,
b
a
a
then  0  0   0
b
b
5.3.4. Multiplication with mixed numbers
5.3.4.1. whole number  mixed number
5.3.4.2. mixed number  whole number
5.3.4.3. rational number  mixed number
5.3.4.4. mixed number  rational number
5.3.4.5. mixed number  mixed number
5.3.5. Division of rational numbers
a
c
5.3.5.1. Definition of division of rational numbers: If
and
are any rational
b
d
c
a c
numbers and
is not zero, then   x if, and only if, x is the unique rational
d
b d
c
a
number such that  x 
d
b
a c a d
c
5.3.5.2. Algorithm for the division of fractions:    where
0
b d b c
d
5.3.6. Estimation and mental math with rational numbers
5.3.6.1. See example 5-16 p. 297
5.3.6.2. See example 5-17 p. 298
5.3.7. Extending the notation of Exponents
5.3.7.1. Definition of a to the mth power: am  a
a

a 

 a , where a is any rational


m _ factors
number and m is any natural number
5.3.7.2. Property: For any non-zero rational number a and any integers m and n, am  an
= am + n
5.3.7.3. Property: For any rational number a such that a  0 and for any integers m and
am
n, n  am  n
a
5.3.7.4. Property: For any rational number a  0 and any integers m and n, (am)n = am · n
m
Property: For any non-zero rational number
a
b
am
a
and any integer m,    m
b
b
5.3.7.6.
Property: For any non-zero rational number
a
b
a
and any integer m,  
b
5.3.7.7.
Properties of exponents:
 am  a
a

a 

 a , where m is a positive integer


5.3.7.5.
m
b
 
a
m
m _ factors


= 1, where a  0
a  1m , where a  0
a0
m
a
a a  amn
am

 am  n , where a  0
n
a
 (am)n = amn
m
am
a
    m , where b  0
b
b
m
 (ab) = am  bm
m
m
a
b
      , where a  0, b  0
b
a
5.3.8. Ongoing Assessment p. 302
5.3.8.1. Home work: 2a, 3a, 3e, 4a, 4c, 5a, 5c, 7a, 8a, 27a, 27c, 28a, 28c, 30a, 30c, 32a,
32b

m
n