section 2 : exponents and their laws

Foundations of Math 9 Updated June 2019

Adrian Herlaar, School District 61 www.mrherlaar.weebly.com

Section 2 : Exponents and their Laws

This book belongs to: Block:

Section Due Date Date Handed In Level of Completion Corrections Made and Understood

𝟐. 𝟏

𝟐. 𝟐

𝟐. 𝟑

𝟐. 𝟒

𝟐. 𝟓

Self-Assessment Rubric

Learning Targets and Self-Evaluation

Learning Target Description Mark

𝟐 − 𝟏 How exponents relate to repeated multiplication

Understanding the effect brackets and negatives have on the given base

𝟐 − 𝟐 Exponents laws during multiplication and division of a common base

Exponent laws in power to a power and zero power situations

Transferring the laws to variable bases

𝟐 − 𝟑 Combined operations with a common base

Simplifying expressions with negative bases to achieve a common base

Category Sub-Category Description

Expert (Extending)

4 Work meets the objectives; is clear, error free, and demonstrates a mastery of the Learning Targets

“You could teach this!”

3.5 Work meets the objectives; is clear, with some minor errors, and demonstrates a clear understanding of the Learning Targets

“Almost Perfect, one little error.”

Apprentice (Proficient)

3 Work almost meets the objectives; contains errors, and demonstrates sound reasoning and thought

concerning the Learning Targets

“Good understanding with a few errors.”

Apprentice (Developing)

2 Work is in progress; contains errors, and demonstrates a partial understanding of the

Learning Targets

“You are on the right track, but key concepts

are missing.”

Novice (Emerging)

1.5 Work does not meet the objectives; frequent errors, and minimal understanding of the Learning

Targets is demonstrated

“You have achieved the bare minimum to meet the learning outcome.”

1 Work does not meet the objectives; there is no or minimal effort, and no understanding of the

Learning Targets

“Learning Outcomes not met at this time.”

Comments: ______________________________________________________________________________

________________________________________________________________________________________

Foundations of Math 9

1 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com

Competency Evaluation

A valuable aspect to the learning process involves self-reflection and efficacy. Research has shown that authentic

self-reflection helps improve performance and effort, and can have a direct impact on the growth mindset of the

individual. In order to grow and be a life-long learner we need to develop the capacity to monitor, evaluate, and

know what and where we need to focus on improvement. Read the following list of Core Competency Outcomes

and reflect on your behaviour, attitude, effort, and actions throughout this unit.

4 3 2 1

I listen during instruction and come ready to ask questions

Personal Responsibility

I am on time for class

I am fully prepared for the class, with all the required supplies

I am fully prepared for Tests

I follow instructions keep my Workbook organized and tidy

I am on task during work blocks

I complete assignments on time

I keep track of my Learning Targets

Self-Regulation

I take ownership over my goals, learning, and behaviour

I can solve problems myself and know when to ask for help

I can persevere in challenging tasks

I am actively engaged in lessons and discussions

I only use my phone for school tasks

Classroom

Responsibility and

Communication

I am focused on the discussion and lessons

I ask questions during the lesson and class

I give my best effort and encourage others to work well

I am polite and communicate questions and concerns with my peers and teacher in a timely manner

I clean up after myself and leave the classroom tidy when I leave

Collaborative Actions

I can work with others to achieve a common goal

I make contributions to my group

I am kind to others, can work collaboratively and build relationships with my peers

I can identify when others need support and provide it

Communication

Skills

I present informative clearly, in an organized way

I ask and respond to simple direct questions

I am an active listener, I support and encourage the speaker

I recognize that there are different points of view and can disagree respectfully

I do not interrupt or speak over others

Overall

Goal for next Unit – refer to the above criteria. Please select (underline/highlight) two areas you want to focus on

Rank yourself on the left of each column: 4 (Excellent), 3 (Good), 2 (Satisfactory), 1 (Needs Improvement)

I will rank your Competency Evaluation on the right half of each column



Section 2.1 - Exponents

Exponents

Exponents are simply the short hand of writing repeated multiplication

Much like multiplication is the same as repeated addition

Example: 2 + 2 + 2 = 3 ∗ 2

But exponents work like this

2 ∗ 2 ∗ 2 = 23

Write these out as repeated multiplication.

Example:

54 = 5 ∗ 5 ∗ 5 ∗ 5

23 = 2 ∗ 2 ∗ 2

42 = 4 ∗ 4

Where it gets tricky is with negative bases, it comes down to how the brackets, if any, are used.

Here we go…

(−2)2 this means that everything inside the brackets is multiplied repeatedly

(−2) ∗ (−2)

This has a profound effect on the final result

A negative number multiplied an even number of times will always finish POSITIVE

So..

(−2)4 = (−2)(−2)(−2)(−2)

= 4 ∗ 4

= 16

Three groups of 2

2 multiplied three times



So when we have an EVEN POWER we can REWRITE the statement without the brackets

as a POSTIVE statement.

Watch this:

(−2)4 = 24

A negative number multiplied an odd number of times will always finish NEGATIVE

So..

(−2)5 = (−2)(−2)(−2)(−2)(−2)

= 4 ∗ 4 ∗ (−2)

= 16 ∗ (−2)

−32

So when we have an ODD POWER we can REWRITE the statement without the brackets

as a NEGATIVE statement.

Watch this:

(−2)5 = −25

Now we have covered when there are brackets

But what about when there are no brackets?

So far we know this…

(−𝑎)𝐸𝑣𝑒𝑛 = 𝑎𝑠𝑎𝑚𝑒 𝑝𝑜𝑤𝑒𝑟

(−𝑎)𝑂𝑑𝑑 = −𝑎𝑠𝑎𝑚𝑒 𝑝𝑜𝑤𝑒𝑟

This is a big deal

This is a big deal



Having a base that looks like: −𝒂

Example:

−2 = (−1)2

So that means that…

−23 = (−1)23

= (−1) ∗ 2 ∗ 2 ∗ 2

−8

−24 = (−1)24

= (−1) ∗ 2 ∗ 2 ∗ 2 ∗ 2

−16

Summary

If the negative is in brackets the result depends on the exponents being odd or even.

(−2)4 = 24

(−2)5 = −25

If there are NO BRACKETS, the answer is ALWAYS NEGATIVE

−25 = (−1)25

−24 = (−1)24

Regardless of the power, even

or odd, if the base is negative

and there are no brackets the

answer is ALWAYS NEGATIVE

Even exponent, the answer is always POSITIVE

Odd exponent, the answer is always NEGATIVE



Section 2.1 – Practice Questions

Write the following expressions as repeated multiplication, use brackets when/where necessary.

1. 25

2. (−3)7

3. (5)2 4. −24

5. (−2)4 6. (−3)5

7. −22 8. −73

9. −(−2)3 10. −(−5)6

For each equation, find the whole number that should be the exponent

11. 8 = 2?

12. 81 = 3?

13. 625 = 5?

14. 64 = 2?

15. 216 = 6?

16. 1024 = 2?



Will the following answers end up positive or negative? Why?

17. −𝑎30

18. (−𝑎)30

19. −(𝑎)30

20. (−𝑎)25

21. (−𝑎)𝐸𝑉𝐸𝑁

22. (−𝑎)𝑂𝐷𝐷

Solve the following.

23. 53

24. 63

25. (−4)3

26. (−3)4

27. −(−2)5

28. −26



Section 2.2 – Multiplication and Division of a Common Base

Multiplication of a Common Base

When we start doing operations with exponents, ask a question…

Do I have a COMMON BASE?

o If the answer is NO, you are done

o If the answer is YES, we can continue

Example:

23 ∗ 24 Do I have a COMMON BASE? YUP! It’s 2

What am I looking at then?

Remember from earlier that: 23 = 2 ∗ 2 ∗ 2 and 24 = 2 ∗ 2 ∗ 2 ∗ 2

So,

23 ∗ 24 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2

What did I do? I ADDED the Exponents!

23 ∗ 24 = 23 + 4 = 27

Example: Simplify the Following

i) 31 ∗ 36 = 31 + 6 = 37 24 ∗ 24 = 24 + 4 = 28

ii) 55 ∗ 57 = 55 + 7 = 512 79 ∗ 712 = 79 + 12 = 721

Multiplication Rule

Must have a COMMON BASE

𝑎𝑚 ∗ 𝑎𝑛 = 𝑎𝑚 + 𝑛



Division of a Common Base

Again, this only works with a COMMON BASE

Example:

37 ÷ 35 well we can re-write that as:

37

35

It’s a fraction and when we have the same number top and bottom we can cancel things out!

37

35=

3 ∗ 3 ∗ 3 ∗ 3 ∗ 3 ∗ 3 ∗ 3

3 ∗ 3 ∗ 3 ∗ 3 ∗ 3=

3 ∗ 3

1= 3 ∗ 3 = 32

In other words:

37

35= 37 − 5 = 32

Example: Simplify the following

i) 125 ÷ 122 = 125 − 2 = 123 68 ÷ 62 = 68 − 2 = 66

ii) 354 ÷ 351 = 354 − 51 = 33 95 ÷ 97 = 95 − 7 = 9−2

Division Rule


𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚 − 𝑛



Multiplication and Division with Negatives

It gets tricky again when we bring negatives back into the fray

We need to make sure we have a COMMON BASE

Things are not always what they seem

Example:

(−3)2 ∗ (−3)3 Do we have a COMMON BASE?

Since they are both in brackets, YES WE DO!

So we can do the same as we did previous:

(−3)2 ∗ (−3)3 = (−3)2 + 3 = (−3)5

Example:

−32 ∗ (−3)3 Do we have a COMMON BASE?

Since they are different with respect to brackets, NO WE DON’T

We need to look at how the brackets will affect the result

Will they end up POSITIVE or NEGATIVE?

−32 ∗ (−3)3

So we can re-write it like this:

−32 ∗ −33

From what we learned previously,

−32 ∗ −33 = (−1)32 ∗ (−1)33

And with some reshuffling, a now COMMON BASE and canceling out:

(−1)(−1)32 ∗ 33 = 32 + 3 = 35

Since there are NO brackets, this

will ALWAYS be negative

Since there are brackets, but the

exponent is ODD, this will END UP being

negative



Example:

−43 ∗ (−4)2 = −43 ∗ 42 = (−1)43 ∗ 42 = (−1)43 + 2 = (−1)45 = −𝟒𝟓

Example:

−53 ∗ (−5)2 ∗ (−5)3 = −53 ∗ 52 ∗ −53 = (−1)53 ∗ 52 ∗ (−1)53 = (−1)(−1)53 + 2 + 3 = 𝟓𝟖

Division yields the same scenario

We have to assess the BRACKET situation

Example:

−55

(−5)2

−55

(−5)2=

−55

52=

(−1)55

52= (−1)55 − 2 = (−1)53 = −𝟓𝟑

Example:

24

−22=

24

(−1)22= (−1)24 − 2 = (−1)22 = −𝟐𝟐

Example:

(−3)5

−33=

−35

(−1)33=

(−1)35

(−1)33= (−1)(−1)35 − 3 = 𝟑𝟐

Since there are NO brackets, this

will ALWAYS be negative

Since there are BRACKETS, and an EVEN

exponent, this will be positive

Always Negative Always Positive

Always Negative Always Negative

Always Positive



Section 2.2 – Practice Questions Simplify the following, leaves answer in Exponential Form.

1. 23 ∗ 24 =

2. 32 ∗ 35 =

3. (−4)2 ∗ (−4)5 = 4. −23 ∗ 22 =

5. −32 ∗ −33 = 6. 24 ∗ 32 ∗ 25 ∗ 36 =

7. −22 ∗ (−2)3 = 8. (−4)1 ∗ (−4)2 ∗ (−43) =

9. 34 ∗ −35 ∗ (−3)2 =

10. (−2)8 ∗ (−2)−3 ∗ (−2)−4 =

11. (−5)6 ∗ (5)4 ∗ (−5)2 ∗ (−5)3 =

12. (−3)4(3)5(−3)2(−3)6 =

13. −23 ∗ 24 ∗ −27 ∗ 23 ∗ 2−12 =

14. 51 ∗ −53 ∗ (−5)7 ∗ 56 ∗ (−5)3 =



Simply the following, leave answer in Exponential Form

15. 27 ÷ 23 = 16.

(−3)10

(−3)2

17. (7)4

(7)1 18. (6)8

(6)8

19. −54

53 20. (−2)6

(−2)−3

21. (−5)8

53

22. 812

8−3

23. 2𝑎+3 ∙ 2𝑎−1 24.

5𝑟+1

5𝑟

25. 3−𝑎+4 ∙ 3𝑎−3

26. 32𝑚

3𝑚−1



Section 2.3 – Power to a Power and Zero Power

Power to a Power

(23)4 means what?

o Well if 23 means: 2 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑖𝑡𝑠𝑒𝑙𝑓 3 𝑡𝑖𝑚𝑒𝑠

o Then (23)4 must mean: 23 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑖𝑡𝑠𝑒𝑙𝑓 4 𝑡𝑖𝑚𝑒𝑠

So,

(23)4 = 23 ∗ 23 ∗ 23 ∗ 23

And we know that when you MULTIPLY a COMMON BASE you ADD the exponents

(23)4 = 23 ∗ 23 ∗ 23 ∗ 23 = 23 + 3 + 3 + 3 = 212

Well remember that repeated addition is just multiplication!

Then a Power to a Power means that we can just MULTIPLY the exponents

(23)4 = 23 ∗ 4 = 212

Zero Power

Follow this logic:

24 = 2 ∗ 2 ∗ 2 ∗ 2

23 = 2 ∗ 2 ∗ 2

22 = 2 ∗ 2

21 = 2

20 =? → 20 = 1

Example:

30 = 1 40 = 1 170 = 1

Watch the negatives!

−40 = (−1)40 = (−1)1 = −1 𝑎𝑛𝑑 −(2)0 = −(1) = −1

At each step I have divided by 2

So when I get to the last one, what is 2 ÷ 2?

Power to a Power Rule


(𝑎𝑚)𝑛 = 𝑎𝑚 ∗ 𝑛




1. Explain (53)4 using repeated multiplication.

Simplify the following, write as repeated multiplication, then exponential form:

2. (23)4

3. (43)3

4. (70)5

5. −(22)3

6. [(−2)2]3 7. [(23)4]5

Will the following answers be positive or negative, why?

8. (−22)3 9. (−22)4

10. ((−2)3)3 11. ((−2)2)3

12. {[(−2)3]2}2 13. {[(−22)3]2}2



Evaluate the following, use BEDMAS, leave answer in exponential form

14. (−2 ∙ 3)2

15. −(2 ∙ 3)2

16. (−2 + 3)4

17. (−2 + 3)5

18. (−6

2)

4

19. ((−2)3

(−2)2)3

Simplify the following.

20. (−7)0

21. −70

22. 30

−30

23. 24

20

24. (212

32 ∗24

313)0

25. (2−5 ∗ 24 ∗ 212 ∗ −24)0



Section 2.4 – Variables as Bases

When it comes to the logic of exponents it doesn’t change when we move from

numbers to variables (unknowns)

Multiplication Rule

𝑥4 ∗ 𝑥2 Do I have a COMMON BASE? YUP!

𝑥4 ∗ 𝑥2 = 𝑥4 + 2 = 𝑥6

Division Rule

𝑟12

𝑟4 Do I have a COMMON BASE? YUP!

𝑟12

𝑟4= 𝑟12 − 4 = 𝑟8

Power to a Power Rule

(𝑘𝟐)𝟑 Remember, we MULTIPLY the EXPONENTS

(𝑘2)3 = 𝑘2 ∗ 3 = 𝑘6

Look out for those negatives, be careful!

−𝑞4 ∗ 𝑞2 = (−1)𝑞4 ∗ 𝑞2 = (−1)𝑞6 = −𝑞6

LOGIC IS THE SAME!!




Leave your answers in exponential form

1. 𝑥2 ∙ 𝑥3 =

2. −𝑟7 ∙ 𝑟4 =

3. 𝑡1 ∙ 𝑡6 =

4. (−𝑡)4 ∙ (−𝑡)7 =

5. 𝑧5

𝑧3 6. 𝑚4

𝑚4

7. −𝑘17

𝑘−3 8. (−𝑟)17

−𝑟−2

9. (−𝑎2)3 = 10. −(𝑐4)2 =

11. −(−𝑐3)7 =

12. ((−𝑟)4)2 =



13. (𝑐−4)−2 =

14. ((−𝑐)−5)−2 =

15. −𝑥2 ∙ (−𝑥)3 ∙ 𝑥4 =

16. −𝑓0 ∙ 𝑓2 ∙ (−𝑓)0 =

17. 𝑛−2 ∙ 𝑛4 =

18. (−𝑔)2 ∙ −𝑔3 ∙ (−𝑔) =

19. −𝑚7

𝑚−4

20. 𝑡−3

𝑡−4

21. (−𝑤)14

−𝑤−4

22. 𝑞0

𝑞−6

23. ((−𝑎)5

(−𝑎)7)−2

24. (𝑗6

𝑗2)4



Section 2.5 – Combined Operations

Take everything we have learned so far and put it all together

1st thing: Do I have a COMMON BASE

2nd thing: Work those NEGATIVES to get to a COMMON BASE

3rd thing: Which RULE(S) do I apply

Example:

26 ∗ 24

22=

26 + 4

22=

210

22= 210−2 = 28

Example:

(𝑥4)6

𝑥9=

𝑥4 ∗ 6

𝑥9=

𝑥24

𝑥9= 𝑥24 − 9 = 𝑥15

Example:

(−3)5 ∗ (−3)4

−32=

−35 ∗ 34

(−1)32=

(−1)35 ∗ 34

(−1)32=

35 + 4

32=

39

32= 39 −2 = 37

Example:

(−2)3 ∗ (−2)2

(−2)4=

(−2)3+2

(−2)4=

(−2)5

(−2)4= (−2)5−4 = (−2)1 = −2

Example:

4𝑟 ∗ 𝑟

𝑟=

4𝑟2

𝑟= 4𝑟2 − 1 = 4𝑟



Example:

15𝑡4 ∗ 𝑡5

3𝑡2=

15𝑡4 + 5

3𝑡2=

15𝑡9

3𝑡2=

15

3∗

𝑡9

𝑡2= 5𝑡9 −2 = 5𝑡7

Example:

(22

33)

2

(24

35)

3

= (22∗2

33∗2) (

24∗3

35∗3) = (

24

36) (

212

315) =

24 ∗ 212

36 ∗ 315=

24+12

36+15=

216

321

Last Thing

If bases are separated by addition or subtraction you can only solve them

The rules do not apply to addition and subtraction!

Be Careful!!!

Example:

24 ∗ 22 + 23 =

24 + 2 + 23 =

26 + 23 =

64 + 8 =

72

Example:

37 ÷ 35 − 32 ∗ 32 =

37− 5 − 32 +2 =

32 − 34 =

9 − 81 =

−72

Can’t use exponent rules when

adding or subtracting bases! All

you can do is solve!




Simplify the following, leave answer in exponential form

1. 27 ∙ 25

28

2. (−3)4 ∙ (−3)8

(−3)10

3. (−7)5 ∙ (−7)4

(−7)7 ∙ (−7)1

4. 54 ∙ (−5)5 ∙ 55

−5 ∙ 53

5. (−3)10 ∙ (−3)0

33 ∙ (−3)3 ∙ 31

6. (−2)5 ∙ 23 ∙ (−2)4

24 ∙ −22 ∙ (−2)3

7. −𝑞−4 ∙ 𝑞7

(−𝑞)2

8. 𝑟3 ∙ 𝑟6

−𝑟−4



9. (−𝑤)7 ∙ −𝑤4 ∙ 𝑤

−𝑤5

10. 𝑗0 ∙ −𝑗0

(−𝑗)0

Simplify and then solve the following.

11. (−2)2 ∙ 24 + (−2)3 ÷ 21

12. (−8)−2 ∙ 84 + (−8)13 ÷ 811

13. (−2)5 + (−2)2

(−2)4

14. (−3)4 − (3)2

(−3)3



Extra Work Space



Answer Key

Section 2.1

1. 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2. (−3)(−3)(−3)(−3)(−3)(−3)(−3) 3. 5 ∗ 5 4. (−1)2 ∗ 2 ∗ 2 ∗ 2

5. (−2)(−2)(−2)(−2) 6. (−3)(−3)(−3)(−3)(−3) 7. (−1)2 ∗ 2 8. (−1)7 ∗ 7 ∗ 7

9. (−1)(−2)(−2)(−2) 10. (−1)(−5)(−5)(−5)(−5)(−5)(−5) 11. 3 12. 4

13. 4 14. 6 15. 3 16. 10

17. 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 18. 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 19. 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 20. 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒

21. 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 22. 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒

23. 125 24. 216 25. −64 26. 81

27. 32 28. −64

Section 2.2

1. 27 2. 37 3. (−4)7 4. (−1)25 5. 35 6. 29 ∙ 38 7. 25 8. 46 9. −311 10. (−2)1

11. −515 12. 317 13. 25 14. −520 15. 24 16. (−3)8 17. 73 18. 60 19. −51 20. (−2)9

21. 55 22. 815 23. 22𝑎+2 24. 51 25. 31 26. 3𝑚+1

Section 2.3

1. 𝑉𝑎𝑟𝑦 2. 212 3. 49 4. 70 5. −26 6. (−2)6 7. 260 8. 𝑁𝑒𝑔 9. 𝑃𝑜𝑠 10. 𝑁𝑒𝑔

11. 𝑃𝑜𝑠 12. 𝑃𝑜𝑠 13. 𝑃𝑜𝑠 14. (−6)2 15. −(6)2 16. 14 17. 15 18. (−3)4 19. (−2)3 20. 1

21. −1 22. −1 23. 24 24. 1 25. 1

Section 2.4

1. 𝑥5 2. −𝑟11 3. 𝑡7 4. (−𝑡)11 5. 𝑧2 6. 1 7. −𝑘20 8. 𝑟19 9. −𝑎6 10. −𝑐8

11. 𝑐21 12. (−𝑟)8 13. 𝑐8 14. (−𝑐)10 15. 𝑥9 16. −𝑓2 17. 𝑛2 18. 𝑔6 19. −𝑚11 20. 𝑡

21. −𝑤18 22. 𝑞6 23. (−𝑎)4 24. 𝑗16

Section 2.5

1. 24 2. (−3)2 3. −7 4. 510 5. −33 6. −23 7. −𝑞

8. −𝑟13 9. −𝑤7 10. −1 11. 60 12. 0 13. −7

4 14. −

8

3

section 2 : exponents and their laws

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