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Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-3
(For help, go to Lesson 2-3.)
Rewrite each expression using exponents.
1.
t•t•t•t•t•t•t
2. (6 – m)(6 – m)(6 – m)
3.
(r + 5)(r + 5)(r + 5)(r + 5)(r + 5)
4. 5 • 5 • 5 • s • s • s
Simplify.
5. –54
6.
(–5)4
7. (–5)0
8.
(–5)–4
Check Skills You’ll Need
8-3
Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-3
Solutions
1. t • t • t • t • t • t • t = t7
2. (6 – m)(6 – m)(6 – m) = (6 – m)3
3. (r + 5)(r + 5)(r + 5)(r + 5)(r + 5) = (r + 5)5
4.
5.
6.
7.
8.
5 • 5 • 5 • s • s • s = 53 • s3 = 53s3
–54 = –(5 • 5 • 5 • 5) = –(25 • 25) = –625
(–5)4 = (–5)(–5)(–5)(–5) = (25)(25) = 625
(–5)0 = 1
(–5)–4 = (– 1 )4
5
= (– 1 )(– 1 )(– 1 )(– 1 )
5
5
= ( 1 )( 1 )
25 25
= 1
625
5
5
8-3
Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-3
Quick Check
Rewrite each expression using each base only once.
a. 73 • 72 = 73 + 2
= 75
b. 44 • 41 • 4–2 = 44 + 1 – 2
= 43
Add exponents of powers with the
same base.
Simplify the sum of the exponents.
Think of 4 + 1 – 2 as 4 + 1 + (–2) to
add the exponents.
Simplify the sum of the exponents.
= 60
Add exponents of powers with the
same base.
Simplify the sum of the exponents.
=1
Use the definition of zero as an exponent.
c. 68 • 6–8 = 68 + (–8)
8-3
Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-3
Simplify each expression.
a.
p2 • p • p5
= p2+1+5
Add exponents of powers
with the same base.
= p8
Simplify.
b. 2q • 3p3 • 4q4 = (2 • 3 • 4)(p3)(q • q 4)
= 24(p3) )(q1• q 4)
= 24(p3) )(q1 + q 4)
= 24p3q5
8-3
Commutative and
Associative Properties of
Multiplication
Multiply the coefficients.
Write q as q1.
Add exponents of powers
with the same base.
Simplify.
Quick Check
Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-3
Simplify (3  10–3)(7  10–5). Write the answer in
scientific notation.
(3 
10–3)(7

10–5) =
(3 • 7)(10–3 • 10–5)
= 21  10–8
= 2.1  101 • 10–8
= 2.1 
101 + (– 8)
= 2.1  10–7
Commutative and
Associative Properties of
Multiplication
Simplify.
Write 21 in scientific
notation.
Add exponents of
powers with the same
base.
Simplify.
8-3
Quick Check
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
(For help, go to Lesson 8-3.)
Rewrite each expression using each base only once.
1. 32 • 32 • 32
2. 23 • 23 • 23 • 23
3. 57 • 57 • 57 • 57
4. 7 • 7 • 7
Simplify.
5. x3 • x3
6. a2 • a2 • a2
7. y–2 • y–2 • y–2
8. n–3 • n–3
Check Skills You’ll Need
8-4
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Solutions
1. 32 • 32 • 32 = 3(2 + 2 + 2) = 36
2. 23 • 23 • 23 • 23 = 2(3 + 3 + 3 + 3) = 212
3. 57 • 57 • 57 • 57 = 5(7 + 7 + 7 + 7) = 528
4. 7 • 7 • 7 = 73
5. x3 • x3 = x(3 + 3) = x6
6. a2 • a2 • a2 = a(2 + 2 + 2) = a6
7. y–2 • y–2 • y–2 = y(–2 + (–2) + (–2)) = y–6 = 16
y
8. n–3 • n–3 = n(–3 + (–3)) = n–6 = 16
n
8-4
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Simplify (a3)4.
(a3)4 = a3 • 4
= a12
Multiply exponents when raising a
power to a power.
Simplify.
Quick Check
8-4
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Simplify b2(b3)–2.
b2(b3)–2 = b2 • b3 • (–2)
Multiply exponents in (b3)–2.
= b2 • b–6
Simplify.
= b2 + (–6)
Add exponents when multiplying
powers of the same base.
= b–4
Simplify.
1
= b4
Write using only positive exponents.
Quick Check
8-4
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Simplify (4x3)2.
(4x3)2 = 42(x3)2
Raise each factor to the second power.
= 42x6
Multiply exponents of a power raised
to a power.
= 16x6
Simplify.
Quick Check
8-4
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Simplify (4xy3)2(x3)–3.
(4xy3)2(x3)–3 = 42x2(y3)2 • (x3)–3
Raise the three factors to the second
power.
= 42 • x2 • y6 • x–9
Multiply exponents of a power raised
to a power.
= 42 • x2 • x–9 • y6
Use the Commutative Property of
Multiplication.
= 42 • x–7 • y6
Add exponents of powers with the
same base.
16y6
= x7
Simplify.
8-4
Quick Check
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Quick Check
An object has a mass of 102 kg. The expression
102 • (3  108)2 describes the amount of resting energy in joules the
object contains. Simplify the expression.
102 • (3  108)2 = 102 • 32 • (108)2
Raise each factor within parentheses
to the second power.
= 102 • 32 • 1016
Simplify (108)2.
= 32 • 102 • 1016
Use the Commutative Property of
Multiplication.
= 32 • 102 + 16
Add exponents of powers with the
same base.
= 9  1018
Simplify.
Write in scientific notation.
8-4
More Multiplication Properties of Exponents
ALGEBRA 1 LESSON 8-4
Simplify each expression.
1.
(x4)5
3. (5a4)3
5.
x(x5y–2)3
x20
2.
125a12
4. (1.5  105)2
(2w–2)4(3w2b–2)3
432
b6w2
x16
y6
2.25  1010
6. (3  10–5)(4  104)2
8-4
4.8  104
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
(For help, go to the Skills Handbook, page 758.)
Write each fraction in simplest form.
5
1. 20
5.
6
15
9. 5xy
15x
125
2. 25
6.
8
30
10. 6y2
60
124
4
3. 100
4.
7. 10
8. 18
11. 3ac
12. 24m
12a
6mn2
35
3x
63
Check Skills You’ll Need
8-5
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
Solutions
1.
5
= 5•1 = 1
20
5•4
4
2.
125
25 • 5
=
=5
25
25 • 1
3.
60
20 • 3
3
=
=
100
20 • 5
5
4.
124
4 • 31
=
= 31
4
4•1
5.
6
3•2 2
=
=
15
3•5 5
6.
8
2•4
4
=
=
30 2 • 15
15
7.
10
5•2
2
=
=
35
5•7
7
8.
18
9•2
2
=
=
63
9•7
7
9.
5xy
5•x•y
y
=
=
15x 5 • 3 • x 3
10. 6y2 = 3 • 2 • y2 = 2y2
3x
3•x
x
11. 3ac = 3 • a • c = c
12a
3•4•a 4
12.
8-5
24m
6•4•m
4
=
=
6mn2
6 • m • n2
n2
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
Simplify each expression.
a.
Subtract exponents when dividing
powers with the same base.
x4
4–9
x
=
9
x
= x–5
Simplify the exponents.
1
Rewrite using positive exponents.
= x5
b.
p3 j –4
p3 – (–3)j
=
–3
6
p j
= p6 j –10
p6
= j10
–4 – 6
Subtract exponents when dividing
powers with the same base.
Simplify.
Rewrite using positive exponents.
Quick Check
8-5
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
Quick Check
A small dog’s heart beats about 64 million beats in a year. If
there are about 530 thousand minutes in a year, what is its average
heart rate in beats per minute?
6.4  107 beats
64 million beats
=
530 thousand min
5.3  105 min
Write in scientific notation.
6.4
 107–5
Subtract exponents when dividing
powers with the same base.
6.4
 102
Simplify the exponent.
= 5.3
= 5.3
1.21  102
= 121
Divide. Round to the nearest hundredth.
Write in standard notation.
The dog’s average heart rate is about 121 beats per minute.
8-5
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
3
y3
Simplify
3
y3
4
4
.
34
= 34
(y )
Raise the numerator and the
denominator to the fourth power.
34
= y 12
Multiply the exponent in the denominator.
81
= y 12
Simplify.
Quick Check
8-5
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
a. Simplify
2
3
–3
=
–3
2
3
3
2
.
3
2
Rewrite using the reciprocal of 3 .
Raise the numerator and the
denominator to the third power.
33
= 23
27
3
= 8 or 3 8
Simplify.
8-5
Division Properties of Exponents
ALGEBRA 1 LESSON 8-5
(continued)
–2
4b
b. Simplify – c
– 4b
c
–2
.
=
– c
4b
2
=
– c
4b
2
Rewrite using the reciprocal of – 4b .
c
Write the fraction with a negative
numerator.
(–c)2
= (4b)2
Raise the numerator and denominator
to the second power.
c2
= 16b2
Simplify.
Quick Check
8-5