date name of lesson fractional exponents part 1 8.5 & 8.6...

Algebra 1 Unit 7 Note Sheets

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Date Name of Lesson

8.1 Apply Exponent Properties Involving Products

8.2 Apply Exponent Properties Involving Quotients

8.3 Define and Use Zero and Negative Exponents

Fractional Exponents Part 1

Fractional Exponents Part 2

8.5 & 8.6 Graph Exponential Functions

8.5 & 8.6 Write Exponential Functions

OC 5.5 Solving Equations Involving Exponents


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8.1 Apply Exponent Properties Involving Products Notes WARM UP Simplify the expression.

1. 𝑥𝑥 + 𝑦𝑦 if 𝑥𝑥 = −2 and 𝑦𝑦 = −6 2. 2𝑥𝑥 − 𝑦𝑦2 if 𝑥𝑥 = −1 and 𝑦𝑦 = 3

Label the following: Exponent, Base and Power Product of Powers Property

Guided Practice 1. 73 ∙ 75 2. 9 ∙ 98 ∙ 92

3. (−5)(−5)6 4. 𝑥𝑥4 ∙ 𝑥𝑥3 Your Turn

5. 32 ∙ 37 6. 5 ∙ 59

7. (−7)2(−7) 8. 𝑥𝑥2 ∙ 𝑥𝑥6 ∙ 𝑥𝑥 Power of a Power Property


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Guided Practice 9. (25)3 10. [(−6)2]5

11. (𝑥𝑥2)4 12. [(𝑦𝑦 + 2)6]2 Your Turn

13. (42)7 14. [(−2)4]5

15. (𝑛𝑛3)6 16. [(𝑚𝑚 + 1)5]4 Power of a Product Property

Guided Practice 17. (24 ∙ 13)8 18. (9𝑥𝑥𝑦𝑦)2

19. (−4𝑧𝑧)2 20. −(4𝑧𝑧)2

21. (2𝑥𝑥3)2 ∙ 𝑥𝑥4 Your Turn

22. (42 ∙ 12)2 23. (−3𝑛𝑛)2

24. (9𝑚𝑚3𝑛𝑛)4 25. 5 ∙ (5𝑥𝑥2)4


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8.2 Apply Exponent Properties Involving Quotients Notes WARM UP Evaluate the expression.

Quotient of Powers Property

Guided Practice

1. 810

84 2.

(−3)9

(−3)3

3. 54∙58

57 4. 1

𝑥𝑥4∙ 𝑥𝑥6

Your Turn

5. 611

65 6.

(−4)9

(−4)2


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7. 94∙93

92 8. 1

𝑦𝑦5∙ 𝑦𝑦8

Power of a Quotient Property

Guided Practice

9. �𝑥𝑥𝑦𝑦�3 10. �− 7

𝑥𝑥�2

Your Turn

11. �𝑎𝑎𝑏𝑏�2 12. �− 5

𝑦𝑦�3

13. �𝑥𝑥2

4𝑦𝑦�2

Guided Practice

14. �4𝑥𝑥2

5𝑦𝑦�3 15. �𝑎𝑎

2

𝑏𝑏�5∙ 12𝑎𝑎2

Your Turn

16. �2𝑠𝑠3𝑡𝑡�3∙ �𝑡𝑡

5

16� 17. �3𝑎𝑎

4

5𝑏𝑏�3

18. �𝑥𝑥3

𝑦𝑦�7∙ 13𝑥𝑥8


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8.3 Define and Use Zero and Negative Exponents Notes WARM UP Simplify the expression. Write your answer using exponents.

Simplify the expression.

Definition of Zero Exponent

Definition of Negative Exponent

Guided Practice 1. 3−2 2. (−7)0

3. �15�−2

4. 0−5

Your Turn

5. �23�0 6. (−8)−2

7. 12−3

8. (−1)0


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Guided Practice 9. 6−4 ∙ 64 10. (4−2)2

11. 13−4

12. 5−1

52

Your Turn

13. 14−3

14. (5−3)−1

15. (−3)5 ∙ (−3)−5 16. 6−2

62

Guided Practice Simplify the expression. Write your answer using only positive exponents.

17. (2𝑥𝑥𝑦𝑦−5)3 18. (2𝑥𝑥)−2𝑦𝑦5

−4𝑥𝑥2𝑦𝑦2

Your Turn

19. 3𝑥𝑥𝑦𝑦−3

9𝑥𝑥3𝑦𝑦


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Fractional Exponents Part 1 Notes WARM UP Simplify each.

1. 2. 3.

4. �𝒂𝒂𝟑𝟑

𝟐𝟐𝟐𝟐�𝟓𝟓 5. �𝒔𝒔

𝟖𝟖

𝟑𝟑𝟑𝟑�𝟑𝟑

Recall the following properties: (𝑥𝑥𝑎𝑎)𝑏𝑏 = 𝑥𝑥𝑎𝑎 ∙ 𝑥𝑥𝑏𝑏 = 𝑥𝑥−𝑎𝑎 = 𝑥𝑥𝑎𝑎−𝑏𝑏 = Perfect Squares – Perfect Cubes – Perfect 4th Powers – Definition of Fractional Exponents

Guided Practice Convert to radical notation. Simplify if possible.

1. 61 2� 2. 101 3�

3. 91 2� 4. 1251 3�


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Your Turn

5. 131 2� 6. 111 3�

7. 161 2� 8. 641 4� Guided Practice Write each expression in exponential form.

9. √2 10. √54

11. �√𝑥𝑥3 �2

12. �√5�3

Your Turn

13. √7 14. √26

15. �√𝑥𝑥4 �3 16. �√43 �

2

Guided Practice Write each expression in radical form.

17. (3𝑥𝑥)−2 5� 18. (7𝑦𝑦)−1.5 Your Turn

19. (6𝑦𝑦)−3 4� 20. (11𝑥𝑥)1.5 Guided Practice Write each expression in exponential form.

21. �√𝑥𝑥3 �2 22. 1

��2𝑦𝑦�5


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Your Turn

23. ��𝑦𝑦5 �3 24. 1

� √7𝑥𝑥3 �2

Guided Practice Simplify.

25. (121)−3 2� 26.

Your Turn

27. (16)3 4�

Fractional Exponents Part 2 Notes WARM UP Evaluate each expression.

1. 50 2. −30 3. �23�−2

4. 22 ∙ 2−4 Simplify each expression.

5. 2x−2y3 6. 3x−3y−2

x2

Guided Practice Use the properties of exponents to simplify.

1. 12−1 2� ∙ 125 2�

2. 64 3� ∙6

61 3�

Your Turn

3. 81 2� ∙ 8−5 2� 4. �3

53� ∙30�

323�


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Guided Practice

5. 71 2� ∙ 71 4�

6. 52 5� ∙ 51 3�

Your Turn

7. 102 3� ∙ 105 6�

8. 33 5� ∙ 31 4�

Guided Practice

9. (74)1 2� 10. (5−6)1 3�

Your Turn

11. (39)2 3�

12. (210)−3 5�

Guided Practice

13. �38

56�12�

14. (−16)−1 2� ∙ (−16)5 2�

Your Turn

15. �76

29�13� 16.(−3)−1 3� ∙ (−3)7 3�


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8.5 & 8.6 Graph Exponential Functions Notes Exponential Function

Guided Practice Graph the function 𝑦𝑦 = 2𝑥𝑥. Identify its domain and range.

1.

2. Graph the function 𝑦𝑦 = 5𝑥𝑥and identify its domain and range


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Your Turn

3. Graph the function 𝑦𝑦 = 3𝑥𝑥 and identify its domain and range Guided Practice

1. Graph the functions 𝑦𝑦 = 13∙ 2𝑥𝑥 and 𝑦𝑦 = – 1

3∙ 2𝑥𝑥. Compare each graph with the graph of 𝑦𝑦 = 2𝑥𝑥

Recall the graph of 𝑦𝑦 = 2𝑥𝑥 from problem 4. Create a table for 𝑦𝑦 = 1

3∙ 2𝑥𝑥

Then plot the points and draw the curve. How does the new curve relate to the given? Create a table for 𝑦𝑦 = −1

3∙ 2𝑥𝑥

Then plot the points and draw the curve. How does the new curve relate to the given?


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Your Turn

2. Graph the functions 𝑦𝑦 = 3 ∙ 2𝑥𝑥 and 𝑦𝑦 = – 3 ∙ 2𝑥𝑥. Compare each graph with the graph of 𝑦𝑦 = 2𝑥𝑥.

Recall the graph of 𝑦𝑦 = 2𝑥𝑥 from problem 4. Create a table for 𝑦𝑦 = 3 ∙ 2𝑥𝑥 Then plot the points and draw the curve. How does the new curve relate to the given? Create a table for 𝑦𝑦 = −3 ∙ 2𝑥𝑥 Then plot the points and draw the curve. How does the new curve relate to the given? Guided Practice

1. Graph the function 𝑦𝑦 = �12�𝑥𝑥 and identify its domain and range.


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Your Turn 2. Graph the function 𝑦𝑦 = (0.4)𝑥𝑥 and identify its domain and range.

Guided Practice

3. Graph the functions 𝑦𝑦 = 3 ∙ �12�𝑥𝑥 and 𝑦𝑦 = −3 ∙ �1

2�𝑥𝑥. Compare each graph with the graph of 𝑦𝑦 = �1

2�𝑥𝑥

Fill out the table for = 3 ∙ �1

2�𝑥𝑥 , then graph.

How does this graph compare with 𝑦𝑦 = �1

2�𝑥𝑥

Fill out the table for = −3 ∙ �1

2�𝑥𝑥 , then graph.

𝑦𝑦 = (0.4)𝑋𝑋 How does this graph compare with 𝑦𝑦 = �1

2�𝑥𝑥


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Your Turn 4. Graph the functions 𝑦𝑦 = 5 ∙ (0.4)𝑥𝑥. Compare graph with the graph of 𝑦𝑦 = (0.4)𝑥𝑥


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8.5 & 8.6 Write Exponential Functions Notes Exponential Function

Guided Practice Write a rule for the function.

1.

2. Your Turn

3.


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4. Guided Practice Tell whether the table represents an exponential function. If so, write a rule for the function.

5.

6. Your Turn

7.

8.

x -2 -1 0 1 2

y -14

- 116

-1 -4 -16


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OC 5.5 Solving Equations Involving Exponents Notes You can apply the properties of equations you already know to solve equations involving exponents. You will also need the following property.

Guided Practice Express each of the following in simplest exponential form:

1. 49 2. 116

3. 1 Your Turn

4. 1000 5. 19 6. 16

Exponential Equations with Equal Bases: If two expressions with the same base are equal then the exponents must also be equal. Guided Practice

5. 2𝑥𝑥 = 25 6. 5𝑥𝑥+3 = 57

7. 16𝑥𝑥 = 164

8. 3(5𝑥𝑥) = 375

9. 3𝑥𝑥−2 + 5 = 32 10. 43𝑥𝑥−2 = 1 Your Turn

11. 4𝑥𝑥 = 64 12. 2𝑥𝑥 = 18


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13. 4𝑥𝑥+2 + 3 = 19 14. 33𝑥𝑥−2 = 81

15. 42𝑥𝑥 = 3212 16. 32𝑥𝑥 = 81

17. 6(5𝑥𝑥) = 750 18. �12�3𝑥𝑥+5

= 128


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Practice Questions: 1. Which of the following best describes the data in the table?

𝑥𝑥 1 2 3 4 𝑦𝑦 3 9 27 81

A. Exponential with a growth factor of 3

B. Linear with a rate of change of 6 C. Quadratic with a second difference of 12 D. none of the above

2. Since the year 2001 the population of community A grows exponentially as illustrated in

the table. The exponential rate of growth is 1.3. What are the units for the rate of growth in the table?

𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 2001 2002 2003 2004 2005 2006 𝑝𝑝𝑦𝑦𝑝𝑝𝑝𝑝𝑝𝑝𝑦𝑦 1200 1560 2028 2636.4 3427.32 4455.52

A. people per year

B. years per people

C. years

D. people

3. Determine which of the following equations represent exponential growth or decay.

Equation 1 Equation 2 Equation 3 Equation 4 𝑦𝑦 = 1.5−𝑥𝑥

𝑦𝑦 = 0.8𝑥𝑥

𝑦𝑦 = 0.5−𝑥𝑥

𝑦𝑦 = 2.7𝑥𝑥

A. Equation 1: Growth

Equation 2: Growth Equation 3: Decay Equation 4: Decay

C. Equation 1: Decay Equation 2: Growth Equation 3: Growth Equation 4: Decay

B. Equation 1: Decay Equation 2: Decay Equation 3: Growth Equation 4: Growth

D. Equation 1: Growth Equation 2: Decay Equation 3: Decay Equation 4: Growth


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4. If 𝑓𝑓(𝑥𝑥) = 3 ∙ 4𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = 3 ∙ 2𝑥𝑥, compare the functions and determine which of the following statements is correct.

A. The x-intercept of 𝑓𝑓(𝑥𝑥) is greater than the x-intercept of 𝑔𝑔(𝑥𝑥).

B. The y-intercept of 𝑓𝑓(𝑥𝑥) is greater than the y-intercept of 𝑔𝑔(𝑥𝑥).

C. The functions increase at the same rate.

D. The functions have the same y-intercept.

5. What is the solution for x in 4𝑥𝑥 = 64 ?

A. 𝑥𝑥 = 16 C. 𝑥𝑥 = 3

B. 𝑥𝑥 = 4 D. 𝑥𝑥 = 2

6. What is the solution for x in 52𝑥𝑥−9 = 125 ?

A. 𝑥𝑥 = 6 C. 𝑥𝑥 = 5

B. 𝑥𝑥 = 4 D. 𝑥𝑥 = 3

7. What is the solution to the system graphed?

A. (2, 4)

B. (4, 2)

C. (1, 2)

D. 𝑛𝑛𝑝𝑝 𝑠𝑠𝑝𝑝𝑝𝑝𝑠𝑠𝑠𝑠𝑠𝑠𝑝𝑝𝑛𝑛

8.

What is the solution for x in the system?

�𝑦𝑦 = 8𝑦𝑦 = 2𝑥𝑥

A. 𝑥𝑥 =

13

C. 𝑥𝑥 = 1

B. 𝑥𝑥 =12

D. 𝑥𝑥 = 3


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9. Which of the following represents the function 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 − 5 ?

A.

C.

B.

D.

10. Which of the following functions is equivalent to the function 𝑓𝑓(𝑥𝑥) = �13�2?

I. 𝑔𝑔(𝑥𝑥) = 3−2

II. 𝑔𝑔(𝑥𝑥) =19

III. 𝑔𝑔(𝑥𝑥) = �23�−1

IV. 𝑔𝑔(𝑥𝑥) = 2(−3)

V. 𝑔𝑔(𝑥𝑥) =16

VI. 𝑔𝑔(𝑥𝑥) = 9−1

A. I, III, V

B. II, IV, VI C. I, II, VI D. III, V, VI


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11. Use the table below to help determine which function has the greatest value as x gets larger and larger.

𝑥𝑥 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 + 3 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥 ℎ(𝑥𝑥) = 𝑥𝑥3 𝑠𝑠(𝑥𝑥) = 3𝑥𝑥

3

4

5

6

A. 𝑓𝑓(𝑥𝑥) has the greatest value as x gets larger and larger.

B. 𝑔𝑔(𝑥𝑥) has the greatest value as x gets larger and larger.

C. ℎ(𝑥𝑥) has the greatest value as x gets larger and larger.

D. 𝑠𝑠(𝑥𝑥) has the greatest value as x gets larger and larger.

12. The maximum height reached by a bouncing ball is given by ℎ(𝑥𝑥) = 10(0.75)𝑥𝑥 where h is

measured in feet and x is the bounce number. Describe the domain of this function and what it means when 𝑥𝑥 = 0.

A. The domain is all real numbers. When the bounce number 𝑥𝑥 = 0, the height h of the ball is 10 𝑓𝑓𝑦𝑦𝑦𝑦𝑠𝑠, which represents its original height of the ball before it is dropped and bounces.

B. The domain is all real numbers. When the bounce number 𝑥𝑥 = 0, the height h of the ball is 7.5 𝑓𝑓𝑦𝑦𝑦𝑦𝑠𝑠, which represents its original height of the ball before it is dropped and bounces.

C. The domain is all nonnegative integers, or 0, 1, 2, 3, … . The domain represents the bounce number x and does not have units. When 𝑥𝑥 = 0 the height h of the ball is 10 𝑓𝑓𝑦𝑦𝑦𝑦𝑠𝑠, which represents its original height of the ball before it is dropped and bounces.

D. The domain is all nonnegative integers, or 0, 1, 2, 3, … . The domain represents the bounce number x and does not have units. When 𝑥𝑥 = 0 the height h of the ball is 7.5 𝑓𝑓𝑦𝑦𝑦𝑦𝑠𝑠, which represents its original height of the ball before it is dropped and bounces.

date name of lesson fractional exponents part 1 8.5 & 8.6...

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