Math9 Q2 W6 laws Of Radicals v3 PDF

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Lesson

Laws of Radicals

6

What I Need to Know In the previous lesson, you learned how to write expressions with rational exponents into radicals and vice-versa. This module may help you understand how radicals exist and how it is being simplified. As you go through this lesson, you will learn to:  

Derive the laws of radicals Apply the laws of exponents to support such laws of radicals.

What I Know Activity 1: Exponential to Radical form and Vice-versa Recall how to write an expression with rational exponents to radicals and vice-versa. This activity will help you attain mastery of your previous lessons. The first items in each test serve as examples. I. Write each expression to radical form. 1.

xy ¿ ¿ ¿

2.

x

3.

y

6.

1/ 2

2/ 3 4 / 5

z

4. (3 x 2 ¿1 / 4 5. 2 x 1/ 3

242 /3 1 /4

7. (ab

¿

8. [(m

¿

1 /3 1/ 4

¿

1/ 3

9. 10. (

4 /5 ¿ ¿ x

2

y ¿1 /4

II.

Write each radical expression to exponential form

1.

2.

√ 5 x= (5 x )

1 /2

32 a ¿ ¿ ¿ √5 ¿

6.

√x n

7.

√ xm n

√ x2

8.

√ b3

9.

4

√8 x6 y4

10.

4

3.

3

4.

4

5.

3

√ax b y n

√ y 4 x2 √b3

Guide Questions: 1. How did you write expressions with rational exponents to radicals? 2. What is the relation between its corresponding exponents to its index and to its radicand? 3. When you write radicals to exponential expression, what are the rules to be considered? Did you see any pattern?

Your goal in this section is to understand how laws of exponents are used as basis to solve radical expressions. Recall the laws of exponents as follows;

n

x y

m

1.

x

2.

x ¿ ¿ ¿

xn

•

x m+n

=

(xy)n

3.

=

n

=

x

mn

4. (

x y

n

¿

xn yn

=

Below are radical expressions, identify what law of exponent is being use in order to simplify it. Illustrative Example 3 3 Solve √7 . Solution:

√ 73 3

=

1

= ( 7 3 ¿3 3 /3

7

= 71 = 7

rewriting radical expression to exponential

applying the laws of exponent

m n

x ¿ ¿

=

x mn

Drill: Follow the example above to solve the following items:

√32

4.

√ 64

5.

1. 4

2.

√3

3.

√

√ √ 64 3

√8

•

40 5

3

√√ x a b

6.

What’s New Answer this activity to further explore the key concepts and the relationships between the laws of exponents to the laws of radicals. Activity 2: Try and Learn Write the following radical expressions to exponential and simplify then indicate what law of exponent is used to support such solutions. Base your answers in the previous 1.

√5¿ 2 ¿

5.

2.

√ a2

6.

√ an

7.

3. 4.

n

√a

•

√3

•

√b n

√8

√√ 8 a b

√b

•

√a n

√a √nb

8.

What Is It From the previous activity, how did you relate the laws of exponents to radicals?

√a

2

For instance, if you simplify

a ¿ = ¿ ¿

=

a

2 /2

α . Similarly,

=

√ an n

= ( a1 /n ¿n

=

an / n = a, by applying the law of exponent ( x m ¿ n = x mn . So in general, if we have n √ a ¿n = a. We consider this as the first law of radical. ¿

√a n

Next, if we have exponent

xy ¿ ¿ ¿

=

n

x y

√a n

n

•

√b n

, this is equal to

, we can write •

√b n

=

√ ab n

a

1 /n

•

b

a 1 /n

1 /n

•

as (

b

1 /n

ab ¿ ¿1 /n

, applying the law of hence,

, this is the second law of radical.

Another point to consider is the expression exponent (

√ n

a b

x n ¿ y

=

xn yn

, we can write

a1/ n b1/ n

√n a √n b

=

a1/ n 1/ n b

as (

a b

¿

, applying the law of 1 /n

hence, (

, therefore,

√n a √n b

=

√ n

a b

this is the third law of radical.

a 1/n ¿ b

=

Lastly, the radical

a

1 /n 1 /m

¿

=

a

1 /mn

√√ a m n

when simplified is this; m n

[by law of exponent ( x ¿

=

√√ a m n

x

mn

= ( ]. Since

therefore

√√a mn

=

√a

mn

is the last law of radical.

√ a ¿1 /m n

a

1 /mn

=( =

√a

mn

,

To summarize the laws of radicals, we have the following;

√an n

1.

√a n

2.

√b n

•

√n a √n b

=

√√a

=

3.

α

=

√ n

=

√ ab n

a b

√a What I Have Learned mn

4.

mn

Activity 3: Let’s Apply! Activity 4: Journal Writing I. Apply the laws of radicals to solve the following items: Make a journal about your learning experiences in this lesson by completing each statement below; √3128 3 3 1. 6. √x 3 √2 I learn that … 2.when √ 8I x… I feel great 3

3

√3 3

7.

I have difficulty 3. 32 √in…

8.

This lesson4.is somewhat… √ 16 x 2

√ 18 3

√ √ 64 2 3

9.

The activities I encountered are… 6 5. √64 I realized that…

•

√ 250 3

√ 48 a10 √4 3 a2 4

10.

I need help when… II. Evaluate the following expressions applying the laws of radicals. 1.

2 (100 x 4 ¿1/Rubrics 3/2

2.

64

3.

√ x • √ x3 x √ 25 Ideas 3

Content & Ideas 4.

√x

2n

for Journal writing

Excellent (5)

are clear & supply of words is 6 n adequate

√ 96 aSentences are well Skills in Sentence 5. Construction with √5 3 a2structured evidence of thoughts relevant to the lessons 5

7

Very Good (4)

Good (3)

Ideas are clear and supply of words is not so adequate

Ideas are not so clear and inadequate supply of words

Sentences are well structured but evidence of thoughts relevant to the lessons are not emphasized

Sentences are not well structured; no evidence of thoughts relevant to the lessons

Grammar

Correct use of grammar at all times

Slight error grammar

in

Erroneous grammar

use

of

Summary/Synthesis/Generalization In this lesson, you have learned the following laws of radicals:

√ an

1.

n

2.

n

3. 4.

√a √n a √n b m n √√ a

α

=

√b n

•

=

√√

√ab n

a b

=

n

=

mn

a

Remember that in solving radicals, we also take into consideration the laws of exponents. With this regard, the laws of radicals and the laws of exponents will go hand in hand in dealing with solutions related to these topics.

Prepared by: Jesusa P. Macas Teacher, Iligan City National High School Evaluated by:

Roxane Mae D. Nacua Teacher I, Division of Gingoog City...