Mathematics 9 Quarter 2 Module 4: Zero and Negative Integral Exponents

Mathematics Quarter 2-Module 4: Zero and Negative Integral Exponents

Mathematics

Quarter 2-Module 4

Zero and Negative Integral Exponents

Week 3

Learning Code – M9AL-IId-1

[illustration: teacher on a laptop screen with students learning at home, with books and speech bubbles]

NOT FOR SALE / GOVERNMENT PROPERTY

GRADE 9 Learning Module for Junior High School Mathematics 2

Quarter 2 – Module 4 – New Normal Math for G9 First Edition 2020 Copyright © 2020

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Published by the Department of Education

Secretary: Leonor Magtolis Briones

Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writers: Rhoda P. Luis – TIII

Cherry Lyn R. Solis – MTI

Editor: Andrea R. Intal – Head Teacher III

Juvy G. Delos Santos – Head Teacher III

Rolando A. Ibayan Jr. – Head Teacher VI

Validators: Remylinda T. Soriano, EPS, Math

Angelita Z. Modesto, PSDS

George B. Borromeo, PSDS

Illustrator: Writers

Layout Artist: Writers

Management Team: Malcolm S. Garma, Regional Director

Genia V. Santos, CLMD Chief

Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator

Maria Magdalena M. Lim, CESO V, Schools Division Superintendent

Aida H. Rondilla, Chief-CID

Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator

GRADE 9 Learning Module for Junior High School Mathematics 3

MODULE 4: Zero and Negative Integral Exponents

In the previous grade level, you have learned about the different laws involving positive integral exponents. You were also taught the importance of this concept in our daily lives. In this module, you will learn how to apply the laws involving positive integral exponents to zero and negative integral exponents. With this, you will appreciate more the concepts of exponents.

LEARNING COMPETENCY

The learner

• applies the laws involving positive integral exponents to zero and negative integral exponents. M9AL-IId-1

WHAT I KNOW

Find out how much you already know about the module. Write the letter that you think is the best answer to each question on a sheet of paper. Answer all items. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.

1. Simplify 4x^0.
   a. x
   b. 4
   c. 1
   d. 4x

   
   

2. 2^-2 is equal to __.
   a. -4
   b. −1/4
   c. 4
   d. 1/4

   
   

3. What is the base in the expression –(–6)^5 ?
   a. 6
   b. -6
   c. -6 x 5
   d. 5

   
   

4. What is the product of x^-5 and 2x^7?
   a. 2x^2
   b. 3x^-2
   c. 3x^2
   d. 2x^12

   
   

5. Which of the following is true?
   a. (x^3)^2 = x^5
   b. 5^3 ● 5^7 = 25^10
   c. x^7/x^10 = 1/x^3
   d. 25x^4/5x^4 = 5x

   
   

6. Simplify the expression 3x^-5.
   a. 1/(3x^5)
   b. 3/x^5
   c. x/3^5
   d. 3^5/x

   
   

7. Write the quotient of 3^4/3^-3 as a single power.
   a. 3^7
   b. 3
   c. 1/3
   d. 3^-1

   
   

8. What rule/s should be applied to simplify the expression (3x^3y^2)^-3?
   i. Quotient Rule
   ii. Power Rule
   iii. Negative Rule
   iv. Product Rule
   a. i & iii
   b. ii & iv
   c. ii & iii
   d. iii & iv

   
   

GRADE 9 Learning Module for Junior High School Mathematics 4

9. Simplify the expressions ((3x^-5 y^2)^0)/((4x^-3 y^2)^-2)
   a. 1
   b. 16y^4/x^6
   c. 3/4
   d. 3x^2/4

   
   

10. When a caterpillar larvae hatches, it weighs only 10^-2 grams. However, each day it can eat 10^4 times its initial body weight. How many grams of food can the larvae eat each day?
    a. 10^2
    b. 10^3
    c. 10^4
    d. 10^6

    
    

WHAT’S IN

Let us recall the different laws involving positive integral exponents, where a and b are not equal to zero, and m, n and x are nonnegative integers.

Rule	Formula
Product Rule	a^m · a^n = a^(m + n)
Power Rule	(a^m)^n = a^(mn)
Power of a Product Rule	(a^m b^n)^x = a^(mx) b^(nx)
Quotient Rule	a^m / a^n = a^(m – n)
Power of a Quotient Rule	(a/b)^m = a^m / b^m

transcriber’s note: the table has no printed header row; “Rule | Formula” is supplied as the header.

Study the examples below.

1. x^2 · x^5 = x^(2 + 5) = x^7
2. (m^3)^4 = m^(3 · 4) = m^12
3. (2x^2)^3 = 2^3 x^(2 · 3) = 8x^6
4. (4/y)^2 = 4^2/y^2 = 16/y^2
5. 5^6/5^4 = 5^(6 − 4) = 5^2 = 25

Try This!

Activity 1: Know Me More

Simplify the following expressions.

1. 3^2 · 3^3
2. (-2)^3
3. (p^3/q^2)^2
4. (x^4)^3
5. (2z)^4
6. 15m^11/3m^7
7. (3x^2y^4)^2
8. x^8/x^6
9. (a^5/a^4)^2
10. (24n^12/3n^8)^2

GRADE 9 Learning Module for Junior High School Mathematics 5

The weight of the bumblebee is represented in scientific notation where the exponent of the base 10 is negative. Exponents can be used in a variety of ways to represent length. Specifically, negative exponents are used to represent how small thing is. But what does it mean when a number is raised to a negative number? How are we going to evaluate a number with a negative exponent?

WHAT’S NEW

Communication, Critical Thinking and Collaboration

[illustration: cartoon avatar of a woman with hair in a bun]

CUTE BATS: DO THEY EXIST?

Last December 2019, reports showed that a coronavirus that specialists had never seen before in humans had started to spread in Wuhan, a large city in the Chinese province of Hubei. It is said that when humans contract a coronavirus, it normally happens due to contact with an infected animal. Some of the most common carriers are bats, though they do not typically transmit coronaviruses directly to humans. It is said that the transmission might occur via an intermediate animal, which will usually, though not always, be a domestic one. And maybe this is one of the reasons why most people now fear this mammal.

[photo: a small bat held in a person’s cupped hands]

But did you know that the world’s smallest bat is the kitti’s hog-nosed bat or also known as bumblebee bat. It weighs only 7 x 10^-2 ounces. It is an incredible tiny creature. The bumblebee bats are found in Thailand and Myanmar, where they occupy limestone caves along rivers. Bumblebee bats are vulnerable and endangered species because of the loss of habitat from human interference.

news.mongabay.com/Bat Conversation International

The weight of the bumblebee is represented in scientific notation where the exponent of the base 10 is negative. Exponents can be used in a variety of ways to represent length. Specifically, negative exponents are used to represent how small thing is. But what does it mean when a number is raised to a negative number? How are we going to evaluate a number with a negative exponent?

WHAT IS IT

Let us analyze what you have read.

1. What is the smallest bat in the world?

   
   
   

2. Where can you find these mammals?

   
   
   

3. How heavy is this mammal?

   
   
   

4. What do you think can you do to protect them?

   
   
   

5. Is there really a reason for us to be afraid of bats?

   
   
   

6. What does it mean when a number is raised to a negative number? How are we going to evaluate a number with a negative exponent?

   
   
   

GRADE 9 Learning Module for Junior High School Mathematics 6

In the previous grade level, quotient rule was discussed. The given exponent in the numerator is always greater than the exponent in the denominator. But what will happen when we will be asked to divide an expression where the base and exponent are the same? For instance, what if we are asked to simplify 10^7/10^7.

If we apply the quotient rule to the fraction 10^7/10^7, we obtain 10^0. But we also know that any nonzero number divided by itself is equal to 1.

That is,

10^7/10^7 = 10^0 and 10^7/10^7 = (10·10·10·10·10·10·10)/(10·10·10·10·10·10·10) = 1

Therefore, 10^0 = 1.

In general, we define

Zero Exponent

a^0 = 1, where a ≠ 0 Any number, excluding zero, with an exponent of zero is equal to 1.

Example 1: Evaluate each expression. a. 6^0 b. (-5)^0 c. -924^0 d. (100 + 200)^0

Solution a. 6^0 = 1 b. (-5)^0 = 1 c. -924^0 = -1 d. (100 + 200)^0 = 300^0 = 1

Let us now consider the fraction 10^3/10^5. In this case, the exponent of the numerator is less than the exponent of the denominator. If we will apply the quotient rule, we will obtain 10^-2. If we will divide the common factors of 10, we will get 1/10^2.

That is,

10^3/10^5 = 10^-2 and 10^3/10^5 = (10·10·10)/(10·10·10·10·10) = 1/10^2

These are the same.

Hence, 10^-2 = 1/10^2

GRADE 9 Learning Module for Junior High School Mathematics 7

In general, we have

Negative Exponents

a^-n = 1/a^n, where a ≠ 0 and n is a counting number.

To show that a^-n = 1/a^n, let us consider a^n · a^-n a^n · a^-n = a^(n + (-n)) Product Rule = a^0 = 1 Zero Exponent

Therefore, a^-n is the reciprocal or the multiplicative inverse of a^n.

Example 2: Transform into expressions with positive exponents only and then evaluate. a. 3^-4 b. (1/6)^-2 c. 1/10^-2 d. 2/3^-2

Solution: a. 3^-4 = 1/3^4 = 1/81 b. (1/6)^-2 = 1/(1/6)^2 = 6^2 = 36 c. 1/10^-2 = 1/(1/10^2) = 10^2 = 100 d. 2/3^-2 = 2/(1/3^2) = 2/(1/9) = 2 · 9 = 18

Note: The quotient rule can be simplified now that we have given meaning to zero and negative integral exponents.

Example 3: Simplify the following and write your answers with positive exponents. a. 9x^0 b. 2x^7 · 2x^-3 c. (3b)^-3 d. 24x^5/(12x^-5)

Solution: a. 9x^0 = 9(1) Zero Exponent = 9 b. 2x^7 · 2x^-3 = 4 · x^(7 + (-3)) Product Rule = 4x^4 c. (3b)^-3 = 1/(3b)^3 Negative Exponent = 1/(3^3 b^3) Power of a Product Rule = 1/(27b^3) d. 24x^5/(12x^-5) = (24/12)(x^(5 − (−5))) Quotient Rule = 2x^10

Example 4: Write each expression with positive exponents only. a. x^-4/y^-2 b. 2x^-3/(3y^-1)

GRADE 9 Learning Module for Junior High School Mathematics 8

Solution: a. x^-4/y^-2 = (1/x^4)/(1/y^2) Negative Exponents = (1/x^4) · (y^2/1) Get the reciprocal of the divisor and then multiply = y^2/x^4

b. 2x^-3/(3y^-1) = (2/x^3)/(3/y) Negative Exponents = (2/x^3) · (y/3) Get the reciprocal of the divisor and then multiply = 2y/(3x^3)

Timesaving Method

Any factor of an expression can be moved from the numerator to the denominator, or from the denominator to the numerator, by changing the sign of its exponents.

Note: Only factors, not terms of the numerator and denominator can be handled this way.

Let us answer again example 4a, solve x^-4/y^-2 using the timesaving method or commonly referred to as “shortcut”.

Since the base x in the numerator has a negative exponent move it to the denominator then change the sign of its exponent. The base y in the denominator has a negative exponent so, move it to the numerator, and change the sign of its exponent.

In this case, x^-4/y^-2 will now become y^2/x^4 when simplified without negative exponents.

Let us have more examples.

Example 5: Simplify and write each expression with positive exponents only. a. 2x^-4/(3y) b. 5a^-5/(6b^-2) c. 4x^3y^-2/(8x^-2y^4)

Solution: a. 2x^-4/(3y) = 2/(3x^4 y) Using the timesaving method, move the factor x^-4 to the denominator and change the sign of its exponent.

WHAT’S MORE

Critical Thinking

b. $\dfrac{5a^{-5}}{6b^{-2}}$ Move the factor a-5 to the denominator and change the sign of its exponent. Also, move b-2 to the numerator and change the sign of its exponent.

Hence, we have $\dfrac{5a^{-5}}{6b^{-2}} = \dfrac{5b^{2}}{6a^{5}}$

c. $\dfrac{4x^{3}y^{-2}}{8x^{-2}y^{4}}$ Using the timesaving method, move the factor y-2 to the denominator and the factor x-2 to the numerator. Do not forget to change the signs of their exponents.

$\dfrac{4x^{3}y^{-2}}{8x^{-2}y^{4}} = \dfrac{4x^{3}x^{2}}{8y^{4}y^{2}}$ Simplify and apply the Product Rule.

$= \dfrac{x^{5}}{2y^{6}}$

Therefore, $\dfrac{4x^{3}y^{-2}}{8x^{-2}y^{4}} = \dfrac{x^{5}}{2y^{6}}$.

To simplify the expression: $\dfrac{4x^{3}y^{-2}}{8x^{-2}y^{4}}$, Apply the Quotient Rule.

$\dfrac{4x^{3}y^{-2}}{8x^{-2}y^{4}} = \dfrac{4}{8}x^{3-(-2)}y^{-2-4}$ Simplify

$= \dfrac{1}{2}x^{5}y^{-6}$ Negative Exponent

$\dfrac{4x^{3}y^{-2}}{8x^{-2}y^{4}} = \dfrac{x^{5}}{2y^{6}}$

In simplifying exponential expressions using the rules, we can say that:

1. There is no recommended sequence to follow. You may use the properties in whatever sequence that is convenient.
2. All rules apply to products and quotients, not to sum and differences. For instance,
   (3x)² = 3²x²
   (3 + x)² is not equal to 3² + x²

It is now your turn. Try the following exercises!

Activity 2

Simplify each.

1. 8⁻²
2. 6⁰
3. 15a⁰
4. (1/3)⁰
5. 100⁰ + 200⁰ + 300⁰
6. 2²⁻¹
7. 4⁻³
8. (2/3)⁻²
9. 8⁻³ / 8⁻⁴
10. 16 · 4⁻²

Activity 3

Simplify each. Express your answers using positive exponents only. Assume that the variables are nonzero constants.

1. (-4b⁻²)⁰
2. 5⁴ / 5⁴
3. x⁻⁸ · x⁵
4. 4h⁻² (–2h⁻⁴)
5. 5x⁰ – 10y⁰
6. (ab²)⁻¹
7. a⁶b⁻²
8. (2x²y⁻³)⁴
9. (9y⁰)²
10. (-7m⁻²n⁻¹)⁻²

Activity 4

Simplify each. Express your answers using positive exponents only. Assume that the variables and denominators are nonzero constants.

1. ((10⁴)/(10⁻⁵))⁰
2. p⁻⁹ / q⁻⁶
3. (g⁴ / g⁻¹)⁵
4. (x² / y³)⁻¹
5. (15n⁻⁴p⁵) / (−3n⁶p⁻⁷)
6. (c⁻²d⁴) / (c⁻⁵d²)

WHAT I HAVE LEARNED

Zero Exponent: a⁰ = 1, where a ≠ 0 Any number, excluding zero, with an exponent of zero is equal to 1.

Negative Exponent: a ≠ 0 and n is a nonnegative integer a⁻ⁿ is the reciprocal or the multiplicative inverse of aⁿ.

Timesaving Method Any factor of an expression can be moved from the numerator to the denominator, or from the denominator to the numerator, by changing the sign of its exponents.

In simplifying exponential expressions using the rules, we can say that:

1. There is no recommended sequence to follow. You may use the properties in whatever sequence that is convenient.
2. All rules apply to products and quotients, not to sum and differences.

WHAT I CAN DO

Critical Thinking

I. Simplify the following expressions with positive exponents only.

1. 1 / 2⁻⁴
2. (2x⁴y⁻⁴)(3x²y⁴)
3. (2p³)⁰(p⁴)
4. (−12x⁵y⁻²) / (8x⁻³y)
5. (−3x⁷y⁻⁴z⁰)³
6. (2r⁻⁵s⁹t⁶)⁻³
7. ((x⁻²)y⁴) / y⁻⁵
8. (10x⁵) / (y⁴z⁻⁷)
9. (a + b)⁻²
10. ((x + y)⁻¹) / (2z⁻⁶)

II. Answer the following problems.

1. A seed on a dandelion flower weighs 10⁻³ grams. A dandelion itself can weigh up to 10³ grams. How many times heavier is a dandelion than its seeds? Express your answer in exponential form.
2. The weight of a newborn baby chicken weighs 3⁻² pounds. If an adult chicken can weigh up to 3⁴ times more than a newborn chicken, how much does an adult chicken weigh?

ASSESSMENT

Write the letter of the correct answer on your answer sheet. If your answer is not among the choices, write the correct answer.

1. Simplify (2r³)(2r⁻²)
   a. 2r b. 2r⁵ c. 4r d. 2r⁶

   
   

2. (x⁻²)⁰ is equal to
   a. -x b. x c. 1 d. x⁰

   
   

3. A number with a negative exponent can be expressed as _____________.
   a. a number base with a positive exponent in the denominator
   b. a negative base number with a positive exponent
   c. a negative base number with a negative exponent
   d. a base number with a negative exponent in the denominator

   
   

4. What is the product of 4x⁵y⁻³ and -3x⁻³y³?
   a. x²y b. -12x² c. 12x² d. x⁸y⁶

   
   

5. Simplify: (2x⁻²)⁰ + (3x²)⁰
   a. 2 b. 3 c. 4 d. 5

   
   

6. Simplify the expressions (2b⁴)⁻⁵ with positive exponents only.
   a. 2/b²⁰ b. 32/x c. 1/(32b²⁰) d. b²⁰/32

   
   

7. Which of the following equations is true?
   a. 2⁻³ = -8 b. 2⁻³ = 1/8 c. 2⁻³ = -1/8 d. 2⁻³ = -6

   
   

8. What rule/s are you going to apply in this expression, ((2x³y⁰) / (−7x⁵y³))⁻³?
   i. Quotient Rule
   ii. Power Rule
   iii. Negative Rule
   iv. Factor Rule
   a. i & iii b. i, ii & iv c. i, ii & iii d. ii, iii & iv

   
   

9. Write the quotient of (12x⁴y⁷z⁰) / (−3x³y⁹z⁻⁵) as a single power.
   a. -4x⁷z⁵ b. 4xy² c. 4x / (y²z⁵) d. -4xz⁵ / y²

   
   

10. Simplify the expressions ((a³b⁻⁴c⁰)⁻²) / ((a²b⁵c³)⁻⁴)
    a. a¹⁴ / (b²⁰c²) b. a⁶b⁸c⁹ c. ab / c³ d. a²b²⁸c¹²

    
    

ADDITIONAL ACTIVITIES

Communication, Critical Thinking

I. Read, analyze and solve each problem.

1. Mr. Gonzales posted a problem in the class. The students were asked to find a number that could replace the question mark.

   
   

   (4x²y⁴)⁰ = x⁷ ● x?

   
   

   Mica says that the question mark should be replaced by 0.
   Richard says that the question mark should be 1.
   Amy says that the question mark should be -7.
   Leni says that the question is impossible to answer.

   
   

   Who is correct and explain why?

   
   

2. Is 5⁻³ = -5³? Explain why or why not.

   
   

II. Complete the Math-Rule Map (MRM) by supplying the missing rules inside the clouds.

[chart: flowchart; Math-Rule Map (MRM)] Top to bottom:

• (−2a³b²c⁰ / 3a²b³c⁷)⁻²
• 2ab⁻¹c⁻⁷ / 3 (raised to the power -2)
• 2⁻²a⁻²b²c¹⁴ / 3⁻²
• 3²b²c¹⁴ / 2²a²
• 9b²c¹⁴ / 4a²

Clouds beside the steps: “Quotient Rule” (given), then blank clouds numbered 1, 2, 3. Values are not labeled on the chart.

III. REFLECT! Viruses like COVID-19 spread at an exponential rate causing epidemics affecting millions of people. How can you help stop the spread of this virus in your community?

E-Search

You may also check the following link for your reference and further learnings on representing quadratic function through table of values and graphs:

https://www.youtube.com/watch?v=j7N-vvjEKIQ https://www.youtube.com/watch?v=P2yeKYke_WE

REFERENCES

Yeo, Joseph, et.al (2017), Mathematics Singapore Math pp. 135 – 141, Rex Bookstore Inc.

Oronce, Orlando A. & Mendoza, Marilyn O. (2015), E-Math 9 pp. 2 – 7, Rex Bookstore Inc.

Boye, Dale E. et.al., Intermediate Algebra pp. 171 – 175, PWS-KENT Publishing Company, Boston, Massachusetts

Mathematics 9 Learner’s Material, Department of Education pp. 233 -237

https://www.softschools.com/math/topics/positive_and_negative_integer_exponents/ https://www.katesmathlessons.com/zero-and-negative-exponents.html https://www.azquotes.com/quotes/topics/exponents.html

Zero and Negative Exponents

https://jenisesexton.files.wordpress.com/2016/02/exponent-activity.pdf https://cdn.kutasoftware.com/Worksheets/Alg1/Graphing%20Quadratic%20Functions.pdf https://centralplainsbatremoval.com/fun-bat-facts-the-bumblebee-bats/ https://en.wikipedia.org/wiki/Kitti%27s_hog-nosed_bat https://braingenie.ck12.org/skills/105553 bitmoji application Coronavirus: Here’s how small the enemy is and how it attacks. Accessed June 2020.

Coronavirus: Here’s how small the enemy is and how it attacks

Coronavirus. University of Texas McLellan Lab. Posted Feb. 19, 2020. https://news.utexas.edu/2020/02/19/breakthrough-in-coronavirus-research-results-in-new-map-to-support-vaccine-design/

LET’S ANALYZE

1. What is 10⁻⁹ using positive exponent?
2. What is its value in fraction form and decimal form?
3. Express the approximate measure of the virus particles in scientific notation in meters.

PROBLEM-BASED LEARNING WORKSHEET

SMALL ENEMY

[illustration: coronavirus particle image]

SARS-CoV-2 is the virus that causes the disease COVID-19. The name “coronavirus” is derived from Latin “corona,” meaning crown or wreath. The spikes adorning the virus’s outer surface gave its original images the look of a corona like that surrounding our sun. Coronaviruses have enveloped virions (virus particles) that measure approximately 120 nm (1 nm = 10⁻⁹ meter) in diameter.

Accessed from: Coronavirus: Here’s how small the enemy is and how it attacks. https://www.ocregister.com/2020/04/10/coronavirus-heres-how-small-the-enemy-is-and-how-it-attacks/

[transcriber’s note: this page is printed upside down; the text below is transcribed in its upright reading order]

ANSWER KEY

WHAT I KNOW

1. B
2. D
3. B
4. A
5. C
6. B
7. A
8. C
9. B
10. A

WHAT’S IN

Activity 1

1. 243
2. -8
3. q⁴/p⁶ [?]
4. x¹² [?]
5. 16⁴ [?]
6. 5m⁴ [?]
7. 9x⁴y⁶ [?]
8. x⁵ [?]
9. a⁵ [?]
10. 16a⁸ [?]

WHAT IS IT

1. Kitti’s hog-nosed or bumblebee bat
2. Thailand and Myanmar
3. 7 x 10⁻³ ounces
   4-6. Answer may vary

WHAT’S MORE

Activity 2

1. 1/64
2. 1
3. 15
4. 1
5. 3
6. 2 [?]
7. 1/64 [?]
8. 9/4 [?]
9. 8
10. 1

Activity 3

1. 1
2. 1
3. 1/x³
4. -8/h⁶ [?]
5. -5

Activity 4

1. 1
2. q⁶/p⁹
3. g²⁵
4. y³/x²
5. -5p¹²/n¹⁰
6. c³d²

WHAT I CAN DO

1. 16
2. 6x⁶
3. p⁴
4. -3x⁸/(2y³)
5. -27x²¹/y¹²
6. r¹⁵/(8s²⁷t¹⁸)
7. y⁹/x²
8. 10x⁵z⁷/y⁴
9. 1/(a + b)²
10. z⁶/(2(x + y))

ASSESSMENT

1. c
2. c
3. a
4. b
5. a
6. c
7. b
8. c
9. d
10. d

ADDITIONAL ACTIVITIES

I.

1. Amy
2. No; 5⁻³ = 1/125 while -5³ = -125

II.

1. Power Rule
2. Negative Exponent Rule
3. Definition of Exponents/Simplify

Problem-Based Learning Worksheet

1. 1/10⁹
2. 1/1000000000 = 0.000000001 [?]
3. 1.2 × 10⁻⁷ meters [?]