8.1 Exponential Properties Involving Products

Chapter 8 – Exponents and Exponential Functions Answer Key

CK-12 Basic Algebra Concepts 1

8.1 Exponential Properties Involving Products

Answers

1. a. base is a

b. exponent is 5

c. the power is the exponent (5)telling us how many times to multiply “a”

d. written as: 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎

2. negative

3. positive

4. negative

5. Order of operations must be applied. In the first expression you are squaring 5 first

then applying the –sign, resulting in an answer of −(5 ∙ 5) = −25. In the second

expression because the (-5) is in brackets you keep the –sign with the 5 when you

multiply it: (−5 ∙ −5) = 25.

6. 22

7. (−3)3

8. 𝑦5

9. (3𝑎)4

10. 45



11. 3𝑥3

12. (−2𝑎)4

13. 63𝑥2𝑦4

14. 1

15. 0

16. 343

17. −36

18. 625

19. 177147

20. 128

21. 4096

22. 64

23. 0.00001

24. −0.216



25. 10077696

26. 4096

27. 576

28. 𝑥6

29. 𝑥9

30. 𝑦15

31. 6𝑦5

32. 60𝑎7

33. 𝑎12

34. 𝑥2𝑦2

35. 81𝑎8𝑏12

36. −32𝑥5𝑦20𝑧10

37. 12𝑥3𝑦5

38. 8𝑥3𝑦5𝑧5



39. 4𝑎6𝑏6

40. −12800𝑥5

41. 64𝑎14

42. 1728𝑥3𝑦3

43. 6𝑥7𝑦6



8.2 Exponential Properties Involving Quotients

Answers

1. 625

2. 1296

3. 729

4. 0.003251536859

5. 𝑎

6. 𝑥4

7. 𝑥5

8. 𝑎5

9. 𝑎2𝑏2

10. 64

11. 625

12. 10.4976

13. 𝑎3𝑏9



14. 𝑥4 𝑦2

15. 3𝑥𝑦

16. 𝑏4

16𝑎8

17. 𝑥6

18. 64𝑎2

𝑏13

19. 3𝑎

20. 3𝑥4

21. 2.0736𝑎24

22. 5𝑥4

4𝑦

23. 𝑥6

𝑦6

24. 3.75𝑎3

𝑏7

25. . 44̅𝑎3𝑏10

26. 3𝑎2𝑐4



8.3 Negative Exponents

Answers

1. 𝑦2

𝑥

2. 1

𝑥4

3. 𝑥4

4. 1

𝑥

5. 2

𝑥2

6. 𝑥2

𝑦3

7. 3

𝑥𝑦

8. 3

𝑥3

9. 𝑎2

𝑏3𝑐

10. 4𝑦3

𝑥

11. 2𝑦3

𝑥2



12. 𝑏2

𝑎2

13. 27𝑏6𝑐9

𝑎6

14. 1

15. 𝑎8

16. 5𝑥−2𝑦

17. 64𝑎−2𝑏13

18. −27

19. 1

20. 64

21. 24

22. 9

23. 121

144

24. 512

25. 4092



26. 41

27. 36

54 or 729

625

28. 9324

29. 367

9

30. 4

35



8.4 Fractional Exponents

Answers

1. √𝑎23

2. 𝑎2

3. √𝑥23

𝑦

4. 𝑧7

𝑥3𝑦5

5. 𝑥√𝑥

√𝑦3

6. 27𝑥3𝑦−1

7. 2𝑎−3𝑏2

8. 1

108𝑥5𝑦−5 𝑜𝑟 .00925926𝑥5𝑦−5

9. 𝑎2 𝑏−10

10. 𝑥−5𝑦4

11. 3𝑥𝑦

12. 3𝑥7𝑦−2



13. 𝑎−2𝑏−3𝑐

14. 𝑥−1𝑦

15. ±64

16. 1

17. 49

18. 8

27 𝑜𝑟 .2962963

19. 1

27𝑜𝑟 .03703704

20. 4

21. 1

2 𝑜𝑟 .5



8.5 Scientific Notation

Answers

1. 310.2

2. 74000

3. 0.00175

4. 0.000029

5. 0.000000000999

6. 1.2 𝑥 105

7. 1.765244 𝑥 106

8. 6.3 𝑥 10

9. 9.654 𝑥 103

10. 6.53937 𝑥 108

11. 1.000000006 𝑥 109

12. 1.2 𝑥 10

13. 2.81 𝑥 10−3



14. 2.7 𝑥 10−8

15. 3 𝑥 10−3

16. 5.6 𝑥 10−5

17. 5.007 𝑥 10−5

18. 9.54 𝑥 10−12



8.6 Scientific Notation with a Calculator

Answers

1. 2.784 × 1018

2. 1.976 × 10−22

3. 4.59 × 10−5

4. 3.68 × 10−6

5. 1.37 × 1015

6. . 629̅̅ ̅̅ ̅̅ × 1017

7. 1.1045 × 10−23

8. 7.35315 × 105

9. 4.17 × 107

10. 3.47 × 10−11



8.7 Exponential Growth

Answers

1. The general equation for an exponential equation is 𝑦 = 𝑎(𝑏)𝑥, where 𝑎 is the starting

value, 𝑏 is the growth factor, 𝑥 is the number of times the growth factor has been

applied and 𝑦 is the solution.

2. In a linear equation we often find a variable with an exponent, while in an exponential

growth equation the variable is always the exponent.

3. The growth factor of an exponential equation must always be greater than 1.

4. False, the form of the equation is correct, but 𝑏 must be greater than 1.

5. The 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 (when 𝑥 = 0) in any exponential growth function will always be

equal to 𝑎 or the starting value, (regardless of what 𝑏 is, when it is raised to the 𝑥

power and 𝑥 = 0 then 𝑏𝑥 = 1 and 𝑎 times 1 is always 𝑎)

6.



7.



8.



9.



10.



11.



12.



13.



14. 𝑦 = 𝑎(𝑏)𝑥

𝑎 = 10 𝑝𝑒𝑜𝑝𝑙𝑒

𝑏 = 10 𝑝𝑒𝑜𝑝𝑙𝑒

𝑐 = 6 𝑤𝑒𝑒𝑘𝑠

If everyone maintains the chain then 10 million people will receive the letter the

sixth week.

15. 𝑦 = 𝑎(𝑏)𝑥

𝑎 = $200.00

𝑏 = 100% + 7.5% = 1.075

𝑐 = 21 − 10 = 11 𝑦𝑒𝑎𝑟𝑠

By her 21st birthday Nadia will have approximately $443.12 in her account.



8.8 Exponential Decay

Answers

1. Exponential decay is when a function decreases from its original value regularly

according to the noted decay factor.

2. “𝑏” in an exponential decay function must always be a fraction between 0 and 1.

3. If 𝑓(𝑥) = 𝑎(𝑏)𝑥 then 𝑓(0) is always equal to a. (𝑏0 will always equal 1 and 𝑎 × 1 is 𝑎)

This means that the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 of an exponential function is always equal to 𝑎 or

the starting value.

4.



5.

6.



7.

8. a) Growth factor is .70.

b) Initial value is 1.

c) 𝑉(65) = 8.54 × 10−11



9. a.

b. 𝑦 = 𝑎(𝑏)𝑥 The initial value is 2,000,000 and the growth factor is ¾ or .75, with 𝑥

= time in days so: 𝑦 = 2,000,000(. 75)𝑥

c. 10 days after ≈112,627

d. 14 days after ≈ 35.636



8.9 Geometric Sequences and Exponential Functions

Answers

1. A geometric sequence is a sequence of numbers in which each number in the sequence

is found by multiplying the previous number by a fixed amount called the common

ratio.

2. a. 𝑎4 = 1 × 23 = 8

b. 𝑎10 = 1 × 29 = 512

c. 𝑎25 = 1 × 224 = 16777216

d. 𝑎60 = 1 × 259 = 576460752303000000 ≈ 5.76 × 1017

3. Yes, If we divide any number by the number preceding it we always get the same

answer: -1. Therefore it is a sequence found by multiplying the previous number by a

common ratio and therefore a geometric sequence.

4. After 3 days, the population will be 81 million.

5. 6 ÷ 8 = .75, 𝑟 = 0.754

6. 4 ÷ 2 = 2, 𝑟 = 2

7. 3 ÷ 9 =1

3, 𝑟 =

1

3

8. −8 ÷ 2 = −4, 𝑟 = −4

9. 2, 6, 18, 54, 162

10. 90, −30, 10, −10

3,

10

9

11. 6, −12, 24, −48, 96



12. 192 ÷ 48 = 4, 𝑟 = 4, 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒: 3, 12, 48, 192, 768

13. 𝑊ℎ𝑒𝑛 𝑟 =1

3, 𝑡ℎ𝑒𝑛: 81, 27, 9, 3, 1 𝑊ℎ𝑒𝑛 𝑟 = −

1

3 𝑡ℎ𝑒𝑛: 81, −27, 9, −3, 1

14. 𝑟 =2

3, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒:

9

4,

3

2, 1,

2

3,

4

9

15. 𝑎6 = 4 × 25 = 128

16. 𝑎4 = −7 ×3

4

3 = 2.953125

17. 𝑎10 = −10 × −39 = 196830

18.

a. 𝑦 =. 8𝑥

b. 𝑦 =. 85 = 0.328 𝑓𝑒𝑒𝑡

19.

a. 37 = 2187 blocks

b. The 6th stack is 243 blocks tall

20.

a. The ball will bounce 11.25 feet high after the third bounce.

b. The ball will bounce 0.2005 feet high after the seventeenth bounce.

21.

a. The rope will stretch 43.2 feet on the third bounce.

b. The rope will stretch 0.4354 feet on the twelfth bounce.

22.

a. After 5 hrs, or ten doublings, the population will be 30,720

b. After 12 doublings, or 6hrs, the population will be apx 123,000



8.10 Applications of Exponential Functions

Answers

1. After six hours there will be approximately 80.8% left.

2. In the year 2020 there will be approximately 1147 bird species left.

3. In the year 2007 there would be approximately 370 stores in Nadia’s chain.

4. In 12 years there will be approximately $833.82 in the account.

5. Approximately 120,481

6. About 3840 bacteria

7. Approximately $13.44, 5 years from now.

8. a.

b. 𝑦 = 7200 ∗ 0.82𝑥

c. $989.63

9. a. 𝑦 = 10,112,620 ∗ 0.995𝑥

b. 9,715,124

c. About 4.25 years (early in 2008)

d. 10,317,426

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8.1 Exponential Properties Involving Products