EXPONENTS
FINAL REVIEW
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Exponents and Division
Simplify. Your answer should contain only positive exponents.
1)
5
4
5
2)
3
3
3
3)
2
2
2
3
4)
2
4
2
2
5)
3
r
3
2
r
6)
7
k
2
4
k
3
7)
10
p
4
6
p
8)
3
b
10
b
3
9)
8
m
3
10
m
3
10)
7
n
3
2
n
5
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11)
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2
n
12)
8
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10
x
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13)
12
x
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9
y
8
14)
14
x
4
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7
6
x
5
y
4
15)
11
u
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17
u
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9
16)
4
y
4
14
y
x
8
17)
12
y
x
4
10
y
x
8
18)
18
x
8
y
8
10
x
3
19)
5
n
8
20
n
8
20)
16
y
x
4
9
x
8
y
2
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Exponents and Division
Simplify. Your answer should contain only positive exponents.
1)
5
4
5
5
3
2)
3
3
3
1
3
2
3)
2
2
2
3
1
2
4)
2
4
2
2
2
2
5)
3
r
3
2
r
3
r
2
2
6)
7
k
2
4
k
3
7
4
k
7)
10
p
4
6
p
5
p
3
3
8)
3
b
10
b
3
3
10
b
2
9)
8
m
3
10
m
3
4
5
10)
7
n
3
2
n
5
7
2
n
2
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12)
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4
5
x
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13)
12
x
3
9
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x
3
3
y
8
14)
14
x
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3
x
15)
11
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17
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17)
12
y
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4
10
y
x
8
6
5
x
4
18)
18
x
8
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10
x
3
9
x
5
y
8
5
19)
5
n
8
20
n
8
1
4
20)
16
y
x
4
9
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2
16
9
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4
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Create your own worksheets like this one with
Infinite Pre-Algebra
. Free trial available at KutaSoftware.com
©
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Exponents and Multiplication
Simplify. Your answer should contain only positive exponents.
1)
4
2
⋅
4
2
2)
4 ⋅
4
2
3)
3
2
⋅
3
2
4)
2 ⋅
2
2
⋅
2
2
5)
2
n
4
⋅
5
n
4
6)
6
r ⋅
5
r
2
7)
2
n
4
⋅
6
n
4
8)
6
k
2
⋅
k
9)
5
b
2
⋅ 8
b 10)
4
x
2
⋅ 3
x
11)
6
x ⋅
2
x
2
12)
6
x ⋅
6
x
3
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⋅
10
u
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5
⋅
8
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v
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14)
9
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y
2
⋅
9
x
5
y
2
15)
6
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3
n
3
⋅
8
m
2
n
3
16)
6
x
2
⋅
6
x
3
y
4
17)
7
u
2
v
5
⋅
9
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v
3
18)
uv ⋅
4
u
v
5
19)
10
x
y
3
⋅
8
x
5
y
3
20)
3
u
4
v
5
⋅
7
u
2
v
3
21)
(
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2
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2
22)
(
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4
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4
23)
(
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3
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4
24)
(
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2
25)
(
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3
26)
(
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Exponents and Multiplication
Simplify. Your answer should contain only positive exponents.
1)
4
2
⋅
4
2
4
4
2)
4 ⋅
4
2
4
3
3)
3
2
⋅
3
2
3
4
4)
2 ⋅
2
2
⋅
2
2
2
5
5)
2
n
4
⋅
5
n
4
10
n
8
6)
6
r ⋅
5
r
2
30
r
3
7)
2
n
4
⋅
6
n
4
12
n
8
8)
6
k
2
⋅
k
6
k
3
9)
5
b
2
⋅ 8
b
40
b
3
10)
4
x
2
⋅ 3
x
12
x
3
11)
6
x ⋅
2
x
2
12
x
3
12)
6
x ⋅
6
x
3
36
x
4
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v
3
⋅
10
u
3
v
5
⋅
8
u
v
3
560
v
11
u
4
14)
9
x
y
2
⋅
9
x
5
y
2
81
x
6
y
4
15)
6
m
3
n
3
⋅
8
m
2
n
3
48
m
5
n
6
16)
6
x
2
⋅
6
x
3
y
4
36
x
5
y
4
17)
7
u
2
v
5
⋅
9
u
v
3
63
u
3
v
8
18)
uv ⋅
4
u
v
5
4
u
2
v
6
19)
10
x
y
3
⋅
8
x
5
y
3
80
x
6
y
6
20)
3
u
4
v
5
⋅
7
u
2
v
3
21
u
6
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8
21)
(
2
x
2
)
2
4
x
4
22)
(
p
4
)
4
p
16
23)
(
k
3
)
4
k
12
24)
(
7
k
)
2
49
k
2
25)
(
x
2
)
3
x
6
26)
(
2
b
2
)
4
16
b
8
-2-
Create your own worksheets like this one with
Infinite Pre-Algebra
. Free trial available at KutaSoftware.com
©
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Powers of Products and Quotients
Simplify. Your answer should contain only positive exponents.
1)
(
3
a
2
)
3
2)
(
2
n
4
)
4
3)
(
3
x
4
)
4
4)
(
6
b
2
)
2
5)
(
7
y
4
)
2
6)
(
3
a
b
4
)
4
7)
(
2
x
4
y
4
)
3
8)
(
5
m
n
3
)
3
9)
(
x
2
y
2
)
2
10)
(
6
y
x
4
)
2
11)
(
u
4
v
3
)
2
12)
(
2
x
4
y
4
)
4
13)
(
3
x
2
⋅
2
x
2
)
2
14)
(
2
p
3
⋅ 2
p
)
2
15)
(
4
n
3
⋅
n
2
)
2
16)
(
3
x ⋅ 2
x
)
2
17)
(
4
x
4
⋅
x
4
)
3
18)
(
4
n
4
⋅
n
)
2
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________
Powers of Products and Quotients
Simplify. Your answer should contain only positive exponents.
1)
(
3
a
2
)
3
27
a
6
2)
(
2
n
4
)
4
16
n
16
3)
(
3
x
4
)
4
81
x
16
4)
(
6
b
2
)
2
36
b
4
5)
(
7
y
4
)
2
49
y
8
6)
(
3
a
b
4
)
4
81
a
4
b
16
7)
(
2
x
4
y
4
)
3
8
x
12
y
12
8)
(
5
m
n
3
)
3
125
m
3
n
9
9)
(
x
2
y
2
)
2
x
4
y
4
10)
(
6
y
x
4
)
2
36
y
2
x
8
11)
(
u
4
v
3
)
2
u
8
v
6
12)
(
2
x
4
y
4
)
4
16
x
16
y
16
13)
(
3
x
2
⋅
2
x
2
)
2
36
x
8
14)
(
2
p
3
⋅ 2
p
)
2
16
p
8
15)
(
4
n
3
⋅
n
2
)
2
16
n
10
16)
(
3
x ⋅ 2
x
)
2
36
x
4
17)
(
4
x
4
⋅
x
4
)
3
64
x
24
18)
(
4
n
4
⋅
n
)
2
16
n
10
Create your own worksheets like this one with
Infinite Pre-Algebra
. Free trial available at KutaSoftware.com
DIVIDING EXPONENTS
Simplify Simplify. Show your work
1.
2
5
3
12
x
12.
43
324
9
63
ba
abba
2.
cd
cd
6
24
3
13.
66
4224
3
64
nm
nmnm
3.
2
6
4
36
m
m
14.

yxyx
yxyx
332
4763
62
89
4.
32
52
7
28
nm
nm
15.

3
2
5342
2
47
ab
baba
5.
2
8
6
54
a
a
16.
52
45
3
2
4
32
yx
yxxy
6.
23
63
4
32
ba
ba
17.
44
2
352
8
43
qp
pqqp
7.
22
54
3
30
qp
qp
18.

2
83
3
2
2
54
10
25
yx
xyyx
8.
2
3
13
52
b
b
19.

2
32
2
3
3
42
6
34
nm
nmnm
9.
22
46
8
36
s
r
sr
20.


baab
baba
4
2
3
42
2
26
26
23
10.
24
58
15
45
yx
yx
11.
23
69
14
35
qp
qp
Answers
1)
3
4x 2)
2
4d
3)
4
9m
4)
2
4n
5)
6
9a
6)
4
8b
7)
32
10 qp
8)
b4
9)
2
9
24
sr
10)
34
3 yx
11)
64
5
2
p
q
12)
ba
2
2
13)
8
14)
65
6 yx
15)
23
7
2
ab
16)
56
6 yx
17)
7
6q
18)
5
2x
19)
88
16 nm
20)
1312
ba
NEGATIVE EXPONENTS
1.
2
2
2.
1
2
3.
0
2
Rewrite with positive exponents then evaluate Rewrite with positive exponents
4.
1
2
19.


3
2
2
3
22.
3
a
5.
2
2
20.
2
5
4
2
23.
42
cx
6.
3
2
21.


3
2
3
9
24.
3
1
7.
92
0
25.
32
1
cb
8.

1
2 26.
2
4
x
9.

2
2 27.
403
xca
10.

1
2
28.
d
c
2
8
11.

2
2
29.
2
1
10
c
a
12.
2
5
1
30.
3
2
0
2
r
13.

3
2
1
31.
22
5 x
14.
2
4
32.
2
2
4
a
15.

2
6
33.
2
4
3
2
b
a
16.

3
10
34.
22
14
3 a
bx
17.
2
9
1
35.
22
322
5
4
b
cx
18.

3
2
1
7
36.
32
4
3
4
2
x
e
ca
Answers
1) 4 2) 2 3) 1
4)
1
2
5)
1
4
6)
1
8
7)
9
8) 2
9) 4
10)
1
2
11)
1
4
12)
25
13)
8
14)
1
16
15)
1
36
16)
1
1000
17)
81
18)
1
49
19)
8
9
20)
1
2
21)
1
3
22)
3
1
a
23)
4
2
c
x
24)
3
x
25)
3
2
c
b
26)
2
4
x
27)
4
3
x
a
28)
2
8
cd
29)
2
10c
a
30)
2
3
r
x
31)
2
25
x
32)
2
16
a
33)
42
8
ab
34)
4
2
9
x
ab
35)
22
3
25
16
x
b
c
36)
3
4
2ec
ax
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 1 Mrs A Tippett
Exponents and Powers
Investigation: Consider the figures.
1. What are the names of these figures?
2. How can we express the dimensions of these figures as repeated multiplication?
3. How can we express the dimensions of these figures using powers?
4. In the expression 7
, the base is , the exponent is and the coefficient is .
5. How are the expressions
(
2
)
and 2
different? Write each as repeated multiplication.
6. Fill in the blanks.
Word Form
Repeated Multiplication
Exponential Form
Standard Form
two cubed
5 × 5 × 5 × 5
6
(1)
Fifth power of
1−
( ) ( )
33− ×−
Ten to the exponent five
7) Use a calculator to determine which power is larger. Use the
key or the ^ key.
a.
0.5
or
0.3
b.
(
2.1
)
or
2.1
c.
4.6
or
(
5.1
)
8) Evaluate for = 3. Remember to show the substitution step.
a. 6
16
b. 3
× 2
c. 3
9
9) Evaluate when = 5 and = 4. Remember to show the substitution step.
a. 4
+
b.
(
2
)
c.
2
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 2 Mrs A Tippett
The Exponent Rules
To multiply powers with the same base, add the exponents.
×
=

To divide powers with the same base, subtract the exponents.
÷
=

To raise a power to a power, multiply the exponents.
(
)
=
×
1) Write as a single power, then evaluate.
a.
4
÷ 4 =
b.
=
c.
3
÷
(
3
)
=
d.
9
×
=
2) Find the value of x.
a. 3
× 3
= 3

b. 5
÷ 5
= 5
c.
(
6
)
= 6

d. 2
× 2
= 2
3) Which is true? Do not use a calculator.
a. 3
+ 3
= 3
b. 3
× 3
= 3

c. 3

÷ 3
= 3
d. 3
÷ 3
= 3
Working with Exponents
4) Complete the chart.
Exponential Form
Repeated Multiplication
Standard Form
(
3
)
÷
(
3
)
125
(
4
)
×
(
4
)
×
(
4
)
(4)
5) The standard forms of
(
2
)
and 2
are not the same. Use repeated multiplication to explain why this
is true.
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 3 Mrs A Tippett
6) Write as a single power then evaluate:
a.
((
4.5
)
)
b.
(
3.2
)
(
3.2
)
c.
(

)
(

)
d.
.
(
.
)
7) Is each statement true or false? Correct the false statements. Do not use calculators.
a. 6
(
2
)
= 48
b.
(

)
÷
(

)
=
c.
(
5
)
÷
(
5
)
= 5
d.
×
=
8) Evaluate for = 2 and = 4.
a. 3
b. 2
5
c.
 +
d.
3
+
Investigate:
1. Write the volume of each prism in expanded form as = × × .
a. = × ×
b. = × ×
2. Write the volume of each prism using exponents instead of repeated multiplication.
a. =
b. =
3. Use exponent laws to simplify your answers.
a. =
b. =
To simplify powers with coefficients & variables, find the power of the coefficient and of each variable.
For example:
(
2
)
=
(
2
)
(
)
(
)
In general:
(
)
=


= 2
×
×

= 16

=


,
0
9) Simplify.
a.
(
5
)
b.
(
4

)
c.

d.
(

)
2
2
2
5
11
5
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 4 Mrs A Tippett
10) Multiple Choice: Simplify the expression
(
7
)
.
A. 7
B.
14
C. 49
D.
49
E. 14
11) Simplify.
a.
(
2
)(

)
b.
(
5
)
()
c.

d.
(

)
(
3
)
(6
)
12) Explain the error in each case and correct it.
a.
(
8
)
= 64
b.
(
5
)(
3
)
= 15
c.
(
3
)
= 9

13) Complete the chart.
Division
Expand and Divide
Exponent Rule
2
÷ 2
×××
×××
= ?
= 2

=
?
3
÷ 3
4
÷ 4
10
÷ 10
÷
a) How do the two answers in each row compare?
b) What is the value of 5
? 7
?
?
c) What is the value of any (non-zero) number raised to the exponent 0?
d) What is the value of 0
?
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 5 Mrs A Tippett
Rule: Any base (except zero) raised to the exponent 0 equals 1.
= 1, 0
14) Evaluate without calculators. Hint: the answers are not all 1!
a.
b.
c.
(
5
)
d.
5
15) Complete the chart.
Division
Expand and Reduce
Exponent Rule
2
÷ 2
2 × 2 × 2
2 × 2 × 2 × 2 × 2
=
1
2
2
2
= 2

= 2

3
÷ 3
5 ÷ 5
10
÷ 10
÷
a) How do the two answers in each row compare?
b) State another way to write 3

, 4

and 1

c) Write a rule for writing a number with a negative exponent.
16) Evaluate without a calculator. Show your work. Leave answers in fraction form.
a. 2

b. 5

c. 3

d. 4

e. 1

Rule: Any base (except zero) raised to a negative exponent equals

=
, 0
the reciprocal of that power but with a positive exponent.
17) Evaluate without calculators. Give exact answers as reduced fractions or integers.
a. 4

b.
(
3
)

c. 2
2

d. 8
÷ 8
× 8
18) Simplify. Write as a single power with a positive exponent.
a.
×

b.
(
)

÷
c.

×
÷

d.

÷
(

)

Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 6 Mrs A Tippett
19) Find the missing value of y.
a.
5
× 5
÷ 5
= 1
b.
(

)

=

c.
5
× 5

=
d.
(

)
(

)
=
20) Evaluate. Leave answers as reduced fractions. Hint: write fractions as division!
a.

+

b.


c.


d.
3
2
2
3
−
−
Rule: Any fractional base raised to a negative exponent equals

=
, 0, 0
the reciprocal of the fraction but with a positive exponent.
Verify: Show that

=
using a calculator and also using exponent rules.
21) Evaluate. Give exact answers as reduced fractions. Hint: watch for common bases!
a.

b.


c.

× 2

d.

22) Decide whether each of the following statements is always true, sometimes true, or never true, and under
what conditions.
a. Two powers in which the exponents
are both −2 have equal values.
b.
(

)
=
(
+
)
for any x
c.
(
)

=
(

)
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 7 Mrs A Tippett
Answers Page 1:
1) square & cube,
2) Square =
44×
, cube
333=××
,
3) .
23
4,3
4) base is
x
, exponent is 6, coefficient is -7
5)
( ) ( )( )( )( )( )
5
5
55
2 22222 32
22 2
x xxxxx x
x xxxxx x
= =
=⋅⋅⋅⋅⋅=
6)
7a)
3
0.5
b.
2
( 2.1)−
c.
7
4.6
8a) 38, b) 1458 c) 234
9a) 116 b. 63 900 c. 150
Answers Page 2:
1a)
6
4 4096=
b)
3
7 343=
c)
2
39=
d)
7
9 4 782 969=
2a)
5 10
5
x
x
+=
=
b)
85
3
x
x
−=
=
c)
4 12
3
x
x
=
=
d)
45
1
x
x
+=
=
3a)
False, there are no exponent rules for sums of powers, use BEDMAS instead
81 243 324+=
b)
34 7
False, When multiplying powers ADD the exponents
33 3×=
c)
10 2 8
False, When dividing powers, SUBTRACT the exponents
333÷=
d) True
4)
Word Form
Repeated Multiplication
Exponential Form
Standard Form
two cubed
222××
3
2
8
Fourth power of 5
5 × 5 × 5 × 5
4
5
625
Sixth power of 1
111111×××××
6
(1)
1
Fifth power of
1−
( )( )( )( )( )
11111−−−−−
( )
5
1−
1−
Negative 3 squared
( ) ( )
33− ×−
( )
2
3−
9
Ten to the exponent five
10 10 10 10 10××××
5
10
100 000
Exponential Form
Repeated Multiplication
Standard Form
(
3
)
÷
(
3
)
( )( )( )( )( )
( )( )
33333
33
−−−−−
−−
( )
3
3 27−=−
( )
3
5−
( )( )( )
555−−−
125
( ) ( )
3
44− ÷−
(
4
)
×
(
4
)
×
(
4
)
(4)
( )
2
4 16
−=
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 8 Mrs A Tippett
5)
( ) ( )( )( )( )
4
4
2 222216
2 12222
− =−−−−=
− =−⋅⋅⋅⋅
Answers pg 3:
6a)
( )
6
4.5 8303.765625−=
b)
( )
9
3.2 35184.37209=
c)
( )
2
7 49−=
d)
1
1.8 1.8=
7a)
6( 8) 48
False
−=−
∴
b)
( )
2
2
aa
True
−=
∴
c)
( )
1
55
False
−=−
∴
d)
66
yy
True
=
∴
8a) -24 b) 12 c) -6 d) 48.5
Investigate 1a)
222V bbb=××
, b)
11 5 5V yyy= ××
2a)
( )
3
2Vb=
b)
23
11 5Vy=××
3a)
3
8Vb=
b)
3
275Vy=
9a)
6 10
25km
b)
12 3 24
64x yz−
c)
8 28
256
81
mn
d)
30 12
pq
Answers pg 4:
10)
86
. 49D pq
11a)
73
2ab−
b)
99
625st−
c)
35
32
81
mp−
d)
7 34
54xyz−
12a)
( 8)( 8) 64− −=
b)
( )( )
2
aa a=
c)
6 12 30 12 30
3 729pq pq=
13)
Page 5 answers:
14)
1
b)
1
9
c) 1 d)
1−
Division
Expand and Divide
Exponent Rule
2
÷ 2
×××
×××
= 1
= 2

=
0
2
3
÷ 3
33
1
33
×
=
×
22 0
33
−
=
4
÷ 4
444
1
444
××
=
××
33 0
44
−
=
10
÷ 10
10 10 10 10
1
10 10 10 10
×××
=
×××
44 0
10 10
−
=
÷
1
xxxxx
xxxxx
⋅⋅⋅⋅
=
⋅⋅⋅⋅
55 0
xx
−
=
Mathematics 9H Exponents Name:_____________________
St Thomas Aquinas Secondary 9 Mrs A Tippett
15)
a) The must be equal
b)
2
2
1
3
3
−
=
,
3
3
1
4
4
−
=
,
4
4
1
1
1
−
=
c) To make the exponent positive take the reciprocal of the BASE.
16)
4
4
11
2
2 16
−
= =
b)
2
2
11
5
5 25
−
= =
c)
3
3
11
3
3 27
−
= =
d)
2
2
11
4
4 16
−
= =
e)
7
7
1
11
1
−
= =
17)
3
11
4 64
=
b)
( )
2
11
9
3
=
−
c)
13
1
44
−=
d)
461 1
1
88
8
−+ −
= =
18)
2
1
t
b)
5
1
c
c)
6
y
d)
6
1
a
Page 6 answers:
19)
7y = −
b)
10y =
c)
1y = −
d)
0y =
20) 6 b) 2 c)
7
8
−
d)
72−
21)
4
3
b)
25
16
c)
1
2
d)
49
9
22a)
22
34
False
−−
≠
∴
b)
( )
2
2
( 3) 3
True
−=
∴
c)
( )
1
x
x
y
y
False
−
=
∴
Division
Expand and Reduce
Exponent Rule
2
÷ 2
2 × 2 × 2
2 × 2 × 2 × 2 × 2
=
1
2
2
2
= 2

= 2

3
÷ 3
4
33 1
333333 3
×
=
×××××
26 4
33
−−
=
5 ÷ 5
2
51
555 5
=
××
13 2
55
−−
=
10
÷ 10
10 10 10 10 1
10 10 10 10 10 10
×××
=
××××
45 1
10 10
−−
=
÷
4
1xxxxx
xxxxxxxxx x
⋅⋅⋅⋅
=
⋅⋅⋅⋅⋅⋅⋅⋅
59 4
xx
−−
=