Quarter 2 Module 5: Laws of Radicals - ZNNHS

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Republic of the Philippines

Department of Education

Regional Office IX, Zamboanga Peninsula

Z est for P rogress

Z eal of P artnership

Quarter 2 – Module 5:

Laws of Radicals

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

Math Module – Grade 9

Alternative Delivery Mode

Quarter 2 – Module 5: Laws of Radicals

First Edition, 2020

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Development Team of the Module Writer: Armil O. Turtor Reviewer: ISMAEL K. YUSOPH

Management Team: SDS: Ma. Liza R. Tabilon, Ed.D, CESO V ASDS: Dr. Ma. Judelyn J. Ramos

ASDS: Dr. Armando P. Gumapon ASDS: Dr. Judith V. Romaguera

CID CHIEF: Lilia E. Abello, Ed.D EPS-LRMS: Evelyn C. Labad EPS-MATH: Ismael K. Yusoph

PSDS: Antonina D. Gallo, Ed.D.

Principal: Daisy Flor J. Romaguera

What I Need to Know

This module is a one-lesson module. It covers key concepts of deriving

the laws of radicals from the laws of rational exponents.

Lesson 1

• Laws of Radicals

In this lesson you will learn to derive the laws of radicals (M9AL-IIf-2).

Learning Objectives

After the lesson, you must be able to:

• derive the radicals from rational expressions.

• find the nth root of a number.

Lesson

1

DERIVING THE LAWS OF

RADICALS FROM RATIONAL

EXPRESSION

What’s In

Give the square of the following numbers.

Number (n) Square of a Number (n2)

1

12=1•1=1

2

3

4

5

6

7

8

9

10

Question:

1 . How did you get the square of a number?

What’s New

Study the figure below.

1 4 9 16 25

Questions

1. What patterns do you notice?

2. Describe the shape formed by the dots.

3. How many dots will the next figure have?

4. What do you call the numbers 1,4, 9, 16, and 25?

5. What is the area of the square having 2 units on its side?

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What is It

The numbers 1, 4, 9, 16, and 25 are called perfect squares.

Observe that 12 = 1, 22 = 4, 32 = 9, 42 = 16, and 52 = 25. If you look at

these numbers geometrically, four square units represent an area of a square

having a side of two units.

Similarly, a square with an area of 16 square units has a side with a

length of four units.

If the pattern is continued, then what is the length of a side of a square

with an area of n square units? To find this number, you must think of a number

whose square is n. This leads you now to the concept of square roots.

Start by finding the square root of 4. Think of a number such that when you square

it, you get 4. From the list of perfect squares above, you can easily say that 22 = 4. Thus,

2 is a square root of 4. In symbols, √4 = 2. Is 2 the only square root of 4? The

answer is no. Note that (-2)2 is also equal to 4. To differentiate the positive square root

from the negative square root, you call the positive square root the principal square

root. The expression refers to

the principal square root of 4. If it is the negative square root of 4 that you want,

then simply write the negative sign in front of √4 , i.e., -√4.

Why is it necessary to emphasize that n must be nonnegative?

What if n is a negative number?

Suppose n = -9. Think of a real number whose square is -9. No real

number will have a square that is negative. The square of a negative number

is positive, and the square of zero is zero, whereas the square of a positive

number is also positive.

The nth root of a number a is a number that when taken as a factor n

times becomes the number a. Radicals represent the roots of a number.

√ 𝑎 𝑛

= √ 𝑎 . 𝑛

Let n be a positive integer greater than 1, and a be a real number.

The principal nth root of a is denoted by , and has the following

√ defining properties:

1. if a > 0, then √ is the positive nth root of a,

2. if a < 0 and n is odd, then √ is the negative nth root of a, and

3.

If a < 0 and n is even, then has no meaning in real numbers.

Expressions like √4, √7√ , 3√𝑥 are examples of radicals.

In the notation , the symbol √ is called radical sign, a is the radicand, n is

the or order, the entire expression is called a radical. As a convention, if no

index appears in a radical, it is understood that the index is 2.

Illustrative Examples

A.√49=7

B.

C.

D. √ not defined in real numbers

From the previous lesson, if m and n are positive integers, then

(𝑎𝑚) = 𝑎𝑚𝑛

If the formula is to hold when m = 1, then we must have

(a 1) = a = a.

1

This equality would mean that is an nth root of a, and we define this as

equivalent to the radical expression .

Let n be a positive integer greater than 1, and let a be a real

number such that is defined in real numbers. Then

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1 √625

1 6

From the definition of a radical, it follows that

Illustrative Examples

1. 625 2 = = 25

1

2.

3. 1 = 6 1 = 1

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6 4 2

4.

The commutative property also holds for rational exponents. It follows that

Illustrative Examples

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( √ 𝑎 𝑛

)

5.

If m and n are positive integers that are relatively prime,

and let a be a real numbers such that is defined in real

numbers, then

We now consider the case when m and n are both positive even integers. Let

a be a real number. We define

Illustrative Examples

Finally, we now consider negative exponents.

Let m and n be positive integers that are relatively prime,

and let a

be a nonzero real number such as that is defined in real

numbers.

Then,

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In particular, when m = n ( both even), we have

Illustrative Examples

What’s More

Activity 1.

Transform each expression to radical form.

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Activity 2. Find the nth root of the following radicals.

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Activity 3. Match items in Column A to the items in Column B. Write the letter

of the correct answer.

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What I Can Do

Do as indicated.

1. Draw rectangles given the following conditions:

2. Formulate a problem using radicals and solve it.

Assessment

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Answer Key

What’s In

Activity 1.

1

1. 10

2

.

0

3. -7

4.

4

5.

undefine

d

6. -3

7

.

-

3

1

10.

undefine

d

9.

- 5

Activity 3.

REFERENCES

Jose-Dilao, Soledad and Bernabe , Julieta G, Intermediate Algebra ,Revised

Edition

Mathematics 9 Learner’s Module Next

Generation Math

http://www.mathisfun.com { HYPERLINK "http://www.mathbitsnotebook.com" } { HYPERLINK "http://www.courses.lumenlearning.com" }

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