ball at time
t
. The acceleration due to gravity is
g
= 32 feet per second per second. If the initial
velocity of the ball is
v
0
= 272 feet per second, find the speed of the ball after
t
= 6 seconds.
48
. A ball is thrown vertically upward. Its velocity
t
seconds after its release is given by the formula
v
=
v
0

gt,
where
v
0
is its initial velocity,
g
is the acceleration due to gravity, and
v
is the velocity of the
ball at time
t
. The acceleration due to gravity is
g
= 32 feet per second per second. If the initial
velocity of the ball is
v
0
= 470 feet per second, find the speed of the ball after
t
= 4 seconds.
49
.
Even
numbers.
Evaluate
the
ex
pression
2
n
for
the
following
values:
i)
n
= 1
ii)
n
= 2
iii)
n
= 3
iv)
n
=

4
v)
n
=

5
vi)Is the result always an even number? Explain.
ii)
n
= 2
iii)
n
= 3
iv)
n
=

4
v)
n
=

5
vi)
Is the result always an odd number?
Explain.
❧
❧
❧
Answers
❧
❧
❧
1.

186
3.

24
5.
7
7.
13
9.
138
11.

134
13.
1
15.

72
17.

2
19.
1
21.
69
23.
0
25.
9
186
CHAPTER 3.
THE FUNDAMENTALS OF ALGEBRA
27.
36
29.

6
31.

71
33.
1
35.
5
37.
46
39.

29
41.
256 feet
43.
110 degrees
45.

409
◦
F
47.
80 feet per second
49.
i)
2
ii)
4
iii)
6
iv)

8
v)

10
vi)
Yes, the result will always be an
even number because 2 will always
be a factor of the product 2
n
.
3.3.
SIMPLIFYING ALGEBRAIC EXPRESSIONS
187
3.3
Simplifying Algebraic Expressions
Recall the commutative and associative properties of multiplication.
The Commutative Property of Multiplication.
If
a
and
b
are any inte
gers, then
a
·
b
=
b
·
a,
or equivalently,
ab
=
ba.
The Associative Property of Multiplication.
If
a
,
b
, and
c
are any inte
gers, then
(
a
·
b
)
·
c
=
a
·
(
b
·
c
)
,
or equivalently,
(
ab
)
c
=
a
(
bc
)
.
The commutative property allows us to change the order of multiplication
without affecting the product or answer. The associative property allows us to
regroup without affecting the product or answer.
You Try It!
EXAMPLE 1.
Simplify: 2(3
x
).
Simplify:

5(7
y
)
Solution.
Use the associative property to regroup, then simplify.
2(3
x
) = (2
·
3)
x
Regrouping with the associative property.
= 6
x
Simplify: 2
·
3 = 6.
Answer:

35
y
The statement 2(3
x
) = 6
x
is an
identity
. That is, the lefthand side and
righthand side of 2(3
x
) = 6
x
are the same
for all values of
x
. Although the
derivation in
Example 1
should be the proof of this statement, it helps the
intuition to check the validity of the statement for one or two values of
x
.
If
x
= 4, then
2(3
x
) = 2(3(
4
))
and
6
x
= 6(
4
)
= 2(12)
= 24
= 24
If
x
=

5, then
2(3
x
) = 2(3(

5
))
and
6
x
= 6(

5
)
= 2(

15)
=

30
=

30
The above calculations show that 2(3
x
) = 6
x
for both
x
= 4 and
x
=

5.
Indeed, the statement 2(3
x
) = 6
x
is true, regardless of what is substituted for
x
.
188
CHAPTER 3.
THE FUNDAMENTALS OF ALGEBRA
You Try It!
EXAMPLE 2.
Simplify: (

3
t
)(

5).
Simplify:
(

8
a
)(5)
Solution.
In essence, we are multiplying three numbers,

3,
t
, and

5, but
the grouping symbols ask us to multiply the

3 and the
t
first. The associative
and commutative properties allow us to change the order and regroup.