Mathematics 9 Quarter 2 Module 6: Writes Expressions with Rational Exponents as Radicals and Vice Versa

Mathematics Quarter 2 – Module 6 Writes Expressions with Rational Exponents as Radicals and Vice Versa

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CO_Q2_Mathematics9 _ Module 6

Copyright and Development Team

Mathematics – Grade 9 Alternative Delivery Mode Quarter 2 – Module 6: Writes Expressions with Rational Exponents as Radicals and Vice Versa First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer: Elizabeth C. Depositar Editors: Andrea R. Intal, Juvy G. Delos Santos, Rolando A. Ibayan Validators: Remylinda T. Soriano, Angelita Z. Modesto, George B. Borromeo Illustrator: Writer Layout Artist: Writer Management Team: Dennis M. Mendoza, Maria Magdalena M. Lim, Aida H. Rondilla, Lucky S. Carpio

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Mathematics Quarter 2 – Module 6 Writes Expressions with Rational Exponents as Radicals and Vice Versa

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Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

Lesson 1: Writes Expressions with Rational Exponents as Radicals and Vice Versa

You have learned how to use exponents to express powers of numbers. In this module, you will learn to write expressions with rational exponents as radicals by applying the laws of exponents.

[illustration: hand pointing with a finger]

What I Need to Know

LEARNING COMPETENCY

The learners will be able to write expressions with rational exponents as radicals and vice versa. (M9AL-IIf-1)

[illustration: hand holding a pencil]

What I Know

Find out how much you already know about the module. Choose the letter of the best answer. Assume all variables are positive. 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. Express $(2x)^{\frac{1}{2}}$ in radical form.
   a. $2\sqrt{x}$
   b. $\sqrt{2x}$
   c. $x\sqrt{2}$
   d. $\sqrt{2x^{\frac{1}{2}}}$

   
   

2. What is the index of the expression $(5a)^{\frac{2}{3}}$ when written in radical form?
   a. 5
   b. 2
   c. $a$
   d. 3

   
   

3. Which of the following is the equivalent of $(\sqrt{2xy})^{3}$ in expression with rational exponent?
   a. $(2xy)^{\frac{2}{3}}$
   b. $(2xy)^{\frac{3}{2}}$
   c. $2(xy)^{\frac{3}{2}}$
   d. $2xy$

   
   

4. Which of the following is the equivalent of $\sqrt[3]{4}$ in expression with rational exponent?
   a. $4^{\frac{1}{3}}$
   b. $3^{4}$
   c. $4^{3}$
   d. $3^{\frac{1}{4}}$

   
   

5. Which of the following is a radical expression?
   a. $\sqrt[3]{2a}$
   b. $ax^2 + by = c$
   c. $x + y = C$
   d. none

   
   

6. In the expression $(2x)^{\frac{1}{2}}$, what is the radicand?
   a. $2\sqrt{x}$
   b. $\sqrt{2x}$
   c. $x\sqrt{2}$
   d. $2x$

   
   

7. Identify the index of the expression $(2xy)^{\frac{3}{5}}$ when written in radical form.
   a. 5
   b. 3
   c. $xy$
   d. 2

   
   

8. Which of the following is the equivalent of $(3\sqrt{2xy})^{3}$ in expression with rational exponent?
   a. $3(2xy)^{\frac{2}{3}}$
   b. $(2xy)^{\frac{3}{2}}$
   c. $27(2xy)^{\frac{3}{2}}$
   d. $2xy$

   
   

9. Which of the following is the equivalent of $4^{1/3}$ in radical form?
   a. $\sqrt[3]{4}$
   b. $\sqrt{4}$
   c. $\sqrt[4]{3}$
   d. $\sqrt{3}$

   
   

10. Which of the following is the equivalent of $\sqrt[3]{5}$ in expression with rational exponent?
    a. $5^{\frac{1}{3}}$
    b. $5^{4}$
    c. $4^{3}$
    d. $3^{\frac{1}{4}}$

    
    

What’s In

Let us recall how to multiply expressions with rational exponent using laws of exponents.

Study the examples below:

1. $x^{2} \cdot x^{\frac{1}{3}} = x^{2+\frac{1}{3}} = x^{\frac{7}{3}}$
2. $a^{\frac{1}{2}} \cdot a^{\frac{3}{2}} = a^{\frac{4}{2}} = a^{2}$
3. $\left(b^{\frac{3}{4}}\right)^{2} = b^{\frac{3}{2}}$

Try this!

Perform the indicated operation and simplify, if needed.

1. $a^{\frac{2}{3}} \cdot a^{\frac{3}{2}}$
2. $b^{0} \cdot b^{\frac{1}{5}}$
3. $\left(y^{\frac{1}{3}}\right)^{2}$
4. $(4xy^{2})^{\frac{1}{2}}$
5. $\left(m^{\frac{3}{4}} n\right)^{-2}$

Evaluate.

1. $\sqrt{25}$ 4. $\sqrt[4]{16}$

   
   

2. $\sqrt{49}$ 5. $\sqrt[4]{625}$

   
   

3. $\sqrt[3]{27}$

   
   

What’s New

WRITING EXPRESSSIONS WITH RATIONAL EXPONENTS AS RADICALS AND VICE VERSA

In the previous lesson, expressions with rational exponents (fractional exponents) are expressions with exponents that are rational. Radicals and expressions with rational exponents are alternate ways of expressing the same thing.

A radical expression is an algebraic expression written in the form $\sqrt[n]{a}$. The symbol $\sqrt{\ }$ is called the radical sign, $a$ is the radicand and $n$ is the index.

[chart: diagram of a radical; labels: index (n), radical sign, radicand (a)]

Convert Expressions with Rational Exponent into Radicals

If we apply the laws of exponents, we can see there are two possible ways to change an expression with fractional exponent into a radical expression.

$a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^{m} = \left(\sqrt[n]{a}\right)^{m}$ or $a^{\frac{m}{n}} = \left(a^{m}\right)^{\frac{1}{n}} = \sqrt[n]{a^{m}}$

All RULES that apply to whole number exponents, also apply to fractional exponents.

Examples:

1. $7^{\frac{2}{3}} = \left(\sqrt[3]{7}\right)^{2} = \sqrt[3]{7^{2}}$
   In the expression with rational (fractional) exponent as $\frac{2}{3}$, the denominator 3 of the rational exponent determines the index and the numerator 2 as exponent of the radicand 7 or radical.

   
   

2. $(xy)^{\frac{2}{3}} = \sqrt[3]{(xy)^{2}} = \sqrt[3]{x^{2}y^{2}}$
   The exponent applies to x and y because of the parentheses.

   
   

3. $8x^{\frac{2}{5}} = 8\sqrt[5]{x^{2}}$
   The exponent applies only to the variable x.

   
   

4. $x^{-\frac{2}{3}} = \frac{1}{x^{\frac{2}{3}}} = \frac{1}{\left(\sqrt[3]{x}\right)^{2}}$ or $\frac{1}{\sqrt[3]{x^{2}}}$

   
   

5. $\left(a^{2} + b^{3}\right)^{\frac{1}{2}} = \sqrt{a^{2} + b^{3}}$

   
   

Convert Radicals to Expressions with Rational Exponent

Any radical expressions in the form $\sqrt[n]{a^{m}}$ can be written as an expression with rational exponent in the form $a^{\frac{m}{n}}$. The index $n$ of the radical will become the denominator and the exponent $m$ of the radicand will become the numerator of the rational exponent.

1. $\sqrt[3]{b} = b^{\frac{1}{3}}$ The index is 3 and the exponent is 1.
2. $\sqrt[3]{2^{4}} = 2^{\frac{4}{3}}$ The index is 3 and the exponent is 4.
3. $\sqrt[4]{2x} = (2x)^{\frac{1}{4}}$ The index is 4 and the exponent is 1.
4. $2\left(\sqrt[3]{y^{4}}\right) = 2y^{\frac{4}{3}}$ The index is 3 and the exponent is 4; the fractional exponent is only for y.

Try this!

A. Express the following in radical form.

1. $5^{\frac{3}{4}}$ __________
2. $(3xy)^{\frac{2}{3}}$ __________
3. $6a^{\frac{3}{5}}$ __________
4. $b^{\frac{7}{4}}$ __________
5. $(x + y)^{\frac{1}{3}}$ __________

B. Express the following with rational exponent.

1. $\sqrt[8]{ab}$ __________
2. $\left(\sqrt{4}\right)^{9}$ __________
3. $\sqrt[5]{m^{7}}$ __________
4. $\left(\sqrt[9]{23}\right)^{7}$ __________
5. $\sqrt[3]{2^{5}}$ __________

CO_Q2_Mathematics9 _ Module 6

What is It

(Communication and Critical Thinking)

WRITING EXPRESSIONS WITH RATIONAL EXPONENTS AS RADICALS AND VICE VERSA

Example1: $4^{\frac{1}{2}} = (\sqrt{4})^{1} = \sqrt{4^{1}} = \sqrt{4}$

Example2: $27^{\frac{1}{3}} = (\sqrt[3]{27})^{1} = \sqrt[3]{27^{1}} = \sqrt[3]{27}$

Example3: $x^{\frac{3}{4}} = (\sqrt[4]{x})^{3} = \sqrt[4]{x^{3}}$

Conclusion:

The _____________ of the fractional exponent becomes the _______ of the radical. The __________ of the fractional exponent becomes the _________ of the expression under the radical symbol OR the exponent of the entire radical.

Your own example:

What’s More

Direction: Match the cards at the top to their answers at the bottom by transforming the given radical form to exponential form and/or exponential form to radical form. Record your answers in the table at the right.

Activity 1: MATCH-UP

Card	Expression
①	$5^{\frac{1}{2}}$
②	$2m^{\frac{1}{4}}$
③	$(3m)^{\frac{2}{3}}$
④	$(5m)^{\frac{1}{2}}$
⑤	$m^{\frac{7}{2}}$
⑥	$-m^{\frac{1}{4}}$
⑦	$(5m)^{\frac{2}{3}}$
⑧	$(3m^{4}n)^{\frac{2}{3}}$
⑨	$(3m^{4}n^{2})^{\frac{3}{2}}$
⑩	$(2mn)^{\frac{2}{5}}$

Answer	Expression
A	$\sqrt{5m}$
B	$\sqrt{5}$
C	$\sqrt{m^{7}}$
D	$\sqrt{27m^{12}n^{6}}$
E	$\sqrt[3]{9m^{2}}$
F	$\sqrt[3]{25m^{2}}$
G	$\sqrt[3]{3m^{8}n^{2}}$
H	$\sqrt[3]{9m^{8}n^{2}}$
I	$\sqrt[5]{2}$
J	$3\sqrt[4]{m}$
K	$-\sqrt[4]{m}$
L	$\sqrt[4]{2m}$
M	$2\sqrt[4]{m}$
N	$\sqrt[4]{-m}$
O	$\sqrt[5]{4m^{2}n^{2}}$

Record your answers:

Card	Answer
①
②
③
④
⑤
⑥
⑦
⑧
⑨
⑩

Activity 2: TRY THIS OUT

Test Yourself!

A. Write each of the following in radical form. Express your answer in simplified form.

1. $50^{\frac{1}{8}}$ __________
2. $16^{\frac{2}{3}}$ __________
3. $89^{\frac{3}{2}}$ __________
4. $35^{\frac{7}{4}}$ __________
5. $78^{\frac{4}{9}}$ __________
6. $x^{\frac{1}{2}}y^{\frac{3}{2}}$ __________
7. $mp^{\frac{1}{2}}$ __________
8. $2a^{\frac{1}{5}}$ __________
9. $12m^{\frac{3}{5}}$ __________
10. $(a+b)^{\frac{3}{5}}$ __________

B. Write each of the following in rational form:

11. $\sqrt[8]{93^{5}}$ __________
12. $(\sqrt{31})^{9}$ __________
13. $\sqrt[5]{22^{7}}$ __________
14. $(\sqrt[9]{63})^{8}$ __________
15. $\sqrt[3]{7^{5}}$ __________
16. $(\sqrt{2ab})^{3}$ __________
17. $(\sqrt[3]{x^{2}y^{3}z^{4}})^{2}$ __________
18. $(\sqrt[7]{5ab^{2}})^{5}$ __________
19. $(\sqrt[3]{3b})^{2}$ __________
20. $\sqrt[3]{x+2}$ __________

What I Have Learned

REMEMBER

If m and n are positive integers that are relatively prime, and let a be a real number such that $\sqrt[n]{a}$ defined in real numbers, then

$a^{\frac{m}{n}} = (\sqrt[n]{a})^{m} = (a^{\frac{1}{n}})^{m}$

What I Can Do

(Communication, Creative and Critical Thinking)

CHECKING YOUR UNDERSTANDING

A. The approximate number of Calories C that an animal needs each day is given by $C = 72m^{\frac{3}{4}}$, where m is the animal’s mass in kilograms. Find the number of Calories that a 625 kg bear needs each day. To solve this, evaluate the equation when m = 625.

Solution:

[illustration: panda beside bamboo]

B. A particular panda consumes 1944 Calories each day. How much does this panda weigh?

Solution:

[illustration: panda beside bamboo]

Assessment

Read each item carefully. Choose the letter of the best answer. Assume that all variables represent positive real numbers.

1. What is the index of the expression $(4a)^{\frac{3}{7}}$ when written in radical form?
   a. 4
   b. 3
   c. 7
   d. 4a

   
   

2. Which of the following is the equivalent of $2\sqrt{x^{3}}$ in expression with rational exponent?
   a. $2x^{\frac{1}{2}}$
   b. $(2x)^{\frac{3}{2}}$
   c. $2x^{\frac{3}{2}}$
   d. $2x^{\frac{2}{3}}$

   
   

3. Which expression is the same as $(2x)^{\frac{1}{4}}$?
   a. $\sqrt[4]{2x}$
   b. $2\sqrt[4]{x}$
   c. $2x^{\frac{1}{4}}$
   d. $\frac{1}{2x}$

   
   

4. Which of the following is the equivalent radical expression of $(a)^{\frac{4}{5}}$?
   a. $\sqrt[5]{a}$
   b. $\sqrt[5]{a^{4}}$
   c. $\sqrt[4]{5}$
   d. $\sqrt[4]{a^{5}}$

   
   

5. Find the value of C when m = 27 and $C = 72m^{\frac{2}{3}}$.
   a. 468
   b. 486
   c. 646
   d. 648

   
   

6. Which of the following is a radical expression?
   a. x + y = C
   b. $\sqrt[5]{3x^{2}}$
   c. $ax^{2} + by = c$
   d. none

   
   

7. The expression $5a^{\frac{3}{2}}$ in radical form is ____.
   a. $5\sqrt{a}$
   b. $\sqrt{2x}$
   c. $a\sqrt{25}$
   d. $5\sqrt{a^{3}}$

   
   

8. What is the index of the expression $(8x)^{\frac{1}{4}}$ when written in radical form?
   a. 8
   b. 1
   c. 4
   d. 8x

   
   

9. Which of the following is the equivalent of $7\sqrt{y^{3}}$ in expression with rational exponent?
   a. $7y^{\frac{2}{3}}$
   b. $(7y)^{\frac{3}{2}}$
   c. $y^{\frac{3}{2}}$
   d. $7y^{\frac{3}{2}}$

   
   

10. Which of the following is the equivalent of $9\sqrt[3]{ab}$ in expression with rational exponent?
    a. $9ab^{3}$
    b. $(9ab)^{\frac{1}{3}}$
    c. $9a^{\frac{1}{3}}b^{\frac{1}{3}}$
    d. $(ab)^{\frac{1}{3}}$

    
    

Additional Activities

REFLECTIVE LEARNING

In today’s world, there are many luxuries that are supposed to make life easier. At times they may, but other times they make thing more complicated. As a student, how will you apply what you learned about simplifying expressions with zero, negative integral and rational exponent in today’s situation? Do you believe in simple living? What does it mean to live simple? Is it good to live a simple life especially in this time of pandemic?

E-Search

To further explore the concept learned today and if it’s possible to connect the internet, you may visit the following links:

• https://www.youtube.com/watch?v=0gicD4STzpg
• https://www.youtube.com/watch?v=TrJUOKLKlsU
• https://www.youtube.com/watch?v=w79rpKCKlFw

Answer Key

[illustration: key icon]

[transcriber’s note: this page is printed upside down. The entries below were read from the inverted image; entries marked [?] or [illegible] are uncertain or unreadable.]

What I Know

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

What’s In

A.

1. [illegible]
2. [illegible]
3. [illegible]
4. [illegible]
5. [illegible]

B.

1. 5 [?]
2. 5 [?]
3. [illegible]
4. [illegible]
5. [illegible]

What’s New

A.

1. [illegible]
2. [illegible]
3. [illegible]
4. [illegible]
5. [illegible]

B.

1. [illegible]
2. [illegible]
3. [illegible]
4. [illegible]
5. [illegible]

What is It

[illegible]

What’s More

Activity 1: MATCH-UP

[transcriber’s note: answers below were checked against the cards on the previous page.]

Card	Answer
1	B
2	M
3	E
4	A
5	C
6	K
7	F
8	H
9	D
10	O

Activity 2: TRY THIS OUT

A.

1	2	3	4	5
[illegible]	[illegible]	[illegible]	[illegible]	[illegible]

6	7	8	9	10
[illegible]	[illegible]	[illegible]	[illegible]	[illegible]

B.

11	12	13	14	15
[illegible]	[illegible]	[illegible]	[illegible]	[illegible]

16	17	18	19	20
[illegible]	[illegible]	[illegible]	[illegible]	[illegible]

Solution for Activity 2: Try it out [illegible]

Answer Key

[transcriber’s note: this page is printed upside down. Entries below were read from the inverted image; the Assessment answers were checked against the item choices.]

What I Can Do

A. Given m = 625 kg. For the Calories given by $C = 72m^{\frac{3}{4}}$, by substitution, $C = 72(625)^{\frac{3}{4}} = 72(\sqrt[4]{625})^{3}$ $C = 72(5)^{3}$ $C = 72(125)$ $C = 9000$ calories

B. Given C = 1944 calories By substitution, $1944 = 72m^{\frac{3}{4}}$ $27 = m^{\frac{3}{4}}$ $3^{3} = (m^{\frac{3}{4}})$ [?] $m^{\frac{1}{4}} = 3$ [?] $m = 81$

Assessment

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

References

• Dilao, S. J., & Bernabe, J. G. (2009). Intermediate Algebra. pp 100-119. Quezon CIty: SD Publicaton.
• Mathematics 9 Learner’s Material, Department of Education
• Oronce, O. A. & Mendoza, M. O. C. (2003). Exploring Mathematics II Intermediate Algebra, Rex Book Store
• DepEd Project Ease Mathematics Module 6 (downloaded from LRMDS)
• Math Time Vol XXIII No. 2 Printed 2019 Grade 9
• https://courses.lumenlearning.com/cuny-hunter-collegealgebra/chapter/1646/
• https://braingenie.ck12.org/skills/106294
• https://cdn.kutasoftware.com/Worksheets/Alg2/Simplifying%20Rational%20Exponents.pdf
• https://www.slideshare.net/jmpalero/mathematics-9-radical-expressions-1
• http://oer2go.org/mods/en-boundless/www.boundless.com/algebra/definition/radical-expression/index.html
• https://www.mathworksheets4kids.com/search/mysearch.php?zoom_query=Write+each+expression+in+radical+form%3A
• https://www.npr.org/2020/06/17/878205853/5-radical-schooling-ideas-for-an-uncertain-fall-and-beyond
• https://www.mesacc.edu/~scotz47781/mat120/notes/exponents/review/review.html
• https://classroomclipart.com/clipart/Clipart/Mathematics.htm

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