Homework 18 Section 4.3
1. a 32 • 32 = 25 • 25 = 25+5 = 210 = 1024
b. 1024 ÷ 256 = 210 ÷ 28 = 210 – 8 = 22 = 4
c. 4096 ÷ 32 = 4 • 1024 ÷ 32 = 22 210 ÷ 25 = 212 ÷ 25 = 27 = 28– 1 = 256 ÷ 2 = 128
2. a. 28 • 27 ÷ 211 = 215 ÷ 211 = 24 = 16
b. (23)5 ÷ 29 = 215 ÷ 29 = 26 = 64
5
3 5
9
d. 8 ÷ 512 = (2 ) ÷ 2 = the answer to b.
f. 256 × 99 = 256 × (100 – 1) = 25600 – 256 = 25244
h. 803 = (8• 10)3 = 83 103 = (23)3 103 = 29 103 = 2 • 23 103 = 2 • 256 • 103 = 512,000
3. Primary Math 6A, page 11:
(15) a. 9
b. 18 c. 9/5
(16) a. 8
b. 9 c. 4 + 8 = 12
(17) g. 47
h. 130 i. 125 – 5 = 120
4. a. (52)3 = 52 • 52 • 52 = (5 • 5) (5 • 5) (5 • 5) = 52 • 3 = 56
b. (52)m = 52 • 52 • • • 52 = (5 • 5) • (5 • 5) • • • (5 • 5) = 52 • m = 52m ]
c. (a2)3 = a2 • a2 • a2 = (a • a) (a • a) (a • a) = a2 • 3 = a6
5. a. 34 54 = (3•3•3•3) •(5•5•5•5) = (3 • 5) (3 • 5) (3 • 5) (3 • 5) = (3 • 5)4 = 154
b. 34 b4 = (3•3•3) •(b•b•b) = (3 • b) (3 • b) (3 • b) = (3 • b)4 = (3b)4
6. a. 2532
c.
!
a 5 (ab)2 (ab 2 )2
a 5a 2b2a 2b 4 a 9b6
a 9 b6
=
=
=
• 4 = a9–4 b6–4 = a5 b2
4
2 2
4 4
4 4
4
b (a )
b a
a b
a b
d. a=2 b=3. To see how to get the answers for parts a) and b) above, substitute the
values a = 2 and b = 3 into the calculation for part c).
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!
! !
7. a.
!
b. 25b2
5 3 • (2 3 • 3)2 •1 2 6 • 32 • 5 3 2 6 32 5 3
1
5 3 • 24 2 •10 0
=
= 3 3 2 = 3 • 3 • 2 = 23 • • 5 = 40/3
3
2
2
2 • (3• 5) • 3 2 • 3 • 5
3
8 •15 • 3
2 3 5
b. The calculation for part b is the same as in part a if you replace 5 by a, 3 by b, and c
by 2.
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! ! !
!
!
c. Let a = 5, b = 3, c = 2
8. a.
!
6 21 •1018 •15 22 (2 • 3)21 • (2 • 5)18 • (3• 5)22 2 21 • 321 • 218 • 518 • 322 • 5 22
=
=
(2 • 3• 5)11 • (2 4 )7
3011 •16 7
211 • 311 • 511 • 2 28
!
!
2 39 • 343 • 5 40
343 • 5 40
= 39 11 11 = 11 11 = 332 • 529
2 •3 •5
3 •5
!
(2 • 3)3n •10 n •1011 • (3• 5)22 (2 • 3)3n • (2 • 5)n •1011 • (3• 5)22
6 3n •10 n+11 •15 22
8b.
=
=
11
n
(3•10)11 • (2 4 )n
30
311 •1011 • 2 4 n
! •16
=
!
!
!
(2 • 3)3n • (2 • 5)n • (3• 5)22 1011
2 3n • 33n • 2 n • 5 n • 322 • 5 22
=
•
11
4n
311!• 2 4 n
1011
! 3 •2
2 3n+n • 5 n • 5 22 33n+22
2 4 n • 5 n+22 33n+22
=
• 11 =
• 11 = 33n+22"11 • 5 n+22 = 33n+11 • 5 n+22
4n
4n
2
3
3
! 2
10. a. 1030 = 1.03 × 103
15600 = 1.56 × 104 345,000,000 = 3.45 × 108
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!
b. Use the distributive property:
7!
7
7
(3.4 × 10 ) + (5.2 × 10 ) = (3.4 + 5.2) × 10 = 8.6 × 107
(6 × 108) + (9.3 × 108) = (6 + 9.3) × 108 = 15.3 × 108 = 1.53 × 109