Mathematics 9 Quarter 2 Module 8: Operations on Radical Expressions

Mathematics

Quarter 2 – Module 8: Operations on Radical Expressions

CO_Q2_Mathematics 9_ Module 8

Mathematics – Grade 9

Alternative Delivery Mode Quarter 2 – Module 8: Operations on Radical Expressions First Edition, 2020

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Mathematics

Quarter 2 – Module 8: Operations on Radical Expressions

9

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.

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Thank you.

Mathematics 9 Quarter 2 – Module 8: Operations on Radical Expressions

Lesson 1: Operations on Radical Expressions

Radical expressions are common in the formulas in businesses to calculate the unknown variables about depreciation, inflation and interests. Engineers also use radicals for measurements and calculations. Auto mechanics used radicals to calculate the car engine’s efficiency. Rather than use a calculator to approximate the values of these radicals, they may express the answer as an exact value in radical form. In order for them to obtain the accurate and exact value in the calculations, they must have the knowledge and skills of the different operations on radical expressions.

[illustration: hand touching a touchscreen]

What I Need to Know

The learners will be able to:

• Perform operations involving radical expressions.
• Solve simple problems involving operations on radicals.

[illustration: hand holding a pencil]

What I Know

Find out how much you already know about the contents of the module. Write the letter of the correct answer in 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. Which of the following is a square root of 196?
   a. 98
   b. 14
   c. 392
   d. 16

   
   

2. Between what two consecutive whole numbers does √31 lie?
   a. 4 and 5
   b. 6 and 7
   c. 5 and 6
   d. 7 and 8

   
   

3. Subtract. 15√5 − 17√5
   a. 32√10
   b. 32√5
   c. −2√5
   d. −2

   
   

4. Find √64.
   a. –8
   b. 32
   c. 8
   d. 4096

   
   

5. Evaluate and perform the operation of −3√20 − √5
   a. −√5
   b. −7√5
   c. −3√15
   d. already simplified

   
   

6. Find the perimeter of a triangle whose side lengths are 15 cm, 8√7 cm, and √112 cm. Give the answer as a radical expression in simplest form.
   a. (15 + 8√7 + √112) cm
   b. (15 + 12√7) cm
   c. (15 + 24√7) cm
   d. 27√7 cm

   
   

7. Simplify the expression √8y + 5√50y − 2√18y.
   a. 13√2y
   b. 21√2y
   c. (√8 + 5√50 − 2√18)√y
   d. 882y

   
   

8. Simplify the expression: 2 / (−4x + 4√5x)
   a. −2x + 2√5x
   b. 1
   c. (−x − √5x) / (2x² − 10x)
   d. (1 − √5) / 12

   
   

9. Simplify the expression: (√18x + 4√8x − √50x).
   a. 13√2x
   b. 6√2x
   c. 72x
   d. (√18x + 4√8 − √50x)√x

   
   

10. Simplify the expression: (√45b + 4√20b − 2√125b).
    a. √5b
    b. 12√5b
    c. 5b
    d. (√45b + 4√20 − 2√125)√b

    
    

11. Get the product in simplest form √2 (√6 + √7).
    a. √12 + √14
    b. 2√6 + 2√7
    c. √26
    d. 2√3 + √14

    
    

12. Write the product in simplest form. √9 (√3 + 2√2)
    a. 9√3 + 18√2
    b. 3√11
    c. √27 + √72
    d. 3√3 + 6√2

    
    

13. The area of a square garden is 85 square meters. Estimate the side length of the garden.
    a. 9 m
    b. 7 m
    c. 11 m
    d. 12 m

    
    

14. Simplify √(200/49).
    a. 20/7
    b. 2/7
    c. 10√2/7
    d. 2√10/7

    
    

15. A square stepping stone in Atlanta’s Centennial Olympic Park measure 4√2 feet on a side. Which of the following is TRUE of the area of the square stone?
    a. The area is a indeterminate
    b. The area is a rational number
    c. The area is an irrational number
    d. The area cannot be determined

    
    

What’s In

Before we proceed, let us review simplifying radicals.

A radical expression is in its simplest form if: (i) there are no perfect nth powers as factors of the radicand, (ii) there are no fractions inside the radical, and (iii) there are no radicals in the denominator. The following laws of radicals can be useful to simplify radical expressions.

For any positive integers m and n and real numbers a and b.

1. (ⁿ√a)ⁿ = a ex. (∛5)³ = 5
2. ⁿ√(ab) = ⁿ√a ⁿ√b ex. ⁵√(2x) = ⁵√2 ⁵√x Product Rule for Radicals
3. ⁿ√(a/b) = ⁿ√a / ⁿ√b ex. ⁷√(11/y) = ⁷√11 / ⁷√y Quotient Rule for Radicals
4. ⁿ√(aᵐ) = (ⁿ√a)ᵐ ex. ∛(7¹⁵) = (⁸√7)¹⁵ [?]
5. ᵐ√(ⁿ√a) = ᵐⁿ√a ex. ∛(⁴√3) = ¹²√3

Let us simplify the following radical expressions.

1. √49
2. ⁴√81
3. ∛(−64n³)
4. ∛(250x⁹)
5. √(∛16)
6. ¹⁰√(625x⁸)
7. √(3/16)
8. ⁵√((−32y⁵)/x²⁰)
9. 9/√3
10. ∛(3/(5a⁴))

[illustration: hands holding a jigsaw puzzle piece]

What’s New

Operations Involving Radicals

Recall that the radicals x√5 and 3√5 are similar. The radicals ⁷√2x and 5y⁷√2x are also similar since they have the same index and same radicand. Radical expressions can be added, subtracted, multiplied or divided. For example, √3 + 5√3 − 2√3 = 4√3 because x + 5x − 2x = 4x is true for any value of x. Because x + 5x − 2x are expressions with like terms while √3 + 5√3 − 2√3 are radicals with same radicand. However, in √2 + 5√3 − 2√3, √2 and √3 cannot be combined because they are not similar. Hence, their sum can only be indicated as √2 + 3√3. The next examples further illustrate the basic operations with radicals.

[illustration: hand holding a leaf]

Addition and Subtraction of Radicals

Rule: Simplify first the radicand, then perform addition and subtraction on the coefficient of the terms with similar radicand.

Examples:

1. 2√3 + 4√3
   Solution: Add the coefficients and annex their common radicand factor
   2√3 + 4√3 = (2 + 4)√3
   = 6√3

   
   

2. 5√2 − √2
   Solution: Subtract the coefficients and annex their common radicand factor
   5√2 − √2 = (5 − 1)√2
   = 4√2

   
   

3. 6√3 + √2 − 9√2 + √3
   Solution:
   6√3 + √2 − 9√2 + √3 = (6√3 + √3) + (√2 − 9√2) Group similar radicals
   = 7√3 − 8√2 Add the coefficients of similar radicals

   
   

4. √m − 8∛m − 7√m − ∛m
   Solution:
   √m − 8∛m − 7√m − ∛m = (√m − 7√m) + (−8∛m − ∛m) Group similar radicals
   = −6√m − 9∛m Add the coefficients of similar radicals

   
   

5. ∛(54x) + 2∛(16x)
   Solution: The radicals are dissimilar but may be similar after simplifying the radicand of each radical.
   ∛(54x) + 2∛(16x) = ∛(27 · 2x) + 2∛(8 · 2x) Factor each radicand
   = 3∛(2x) + 2(2∛(2x)) Simplify
   = 3∛(2x) + 4∛(2x) Multiply
   = 7∛(2x) Add the coefficients

   
   

6. 2√12 − √48 + 3√27
   Solution:
   2√12 − √48 + 3√27 = 2√(4 · 3) − √(16 · 3) + 3√(9 · 3) Simplify the radicand if possible
   = 2(2√3) − (4√3) + 3(3√3) Simplify
   = 4√3 − 4√3 + 9√3 Multiply
   = 9√3 Combine the coefficients

   
   

7. √20/2 + √(5/4)

   
   

Solution:

√20/2 + √(5/4) = √20/2 + √5/√4 Distribute the radical into numerator and denominator

= √(4·5)/2 + √5/√4 Simplify the radicand

= 2√5/2 + √5/2 Simplify

= (2√5 + √5)/2

= 3√5/2 or (3/2)√5 Add the fractions

8. 10√(49/2) − √50

Solution:

10√(49/2) − √50 = 10(√49/√2) − √(25·2) Distribute the radical into numerator and denominator

= 10(7/√2) − 5√2 Simplify

= 10((7/√2)·(√2/√2)) − 5√2 Rationalize since there is a radical in the denominator

= 10(7√2/2) − 5√2 Get the LCD. LCD is 2

= 70√2/2 − 10√2/2 Write as similar fractions

= 60√2/2 Subtract

= 30√2 Simplify

9. √(2/3) + √(3/2)

Solution:

√(2/3) + √(3/2) = √2/√3 + √3/√2 Distribute the radical into numerator and denominator

= (√2/√3)·(√3/√3) + (√3/√2)·(√2/√2) Rationalize since there is a radical in the denominator

= √6/3 + √6/2 Get the LCD. LCD is 6

= 2√6/6 + 3√6/6 Write as similar fractions

= 5√6/6 Add

10. Find the perimeter of a triangle with sides measure 6√10 cm, 5√10 cm and 11√10 cm.

Solution: To find the perimeter of a triangle, add the measures of the three sides.

P = a + b + c

P = 6√10 + 5√10 + 11√10

P = 22√10 cm

The perimeter of the triangle is 22√10 cm.

Multiplication of Radicals

Rule:

In multiplying radical expressions, we consider the following:

• The indices are the same.

Apply the product rule for radicals, ⁿ√a · ⁿ√b = ⁿ√(ab), and if necessary, simplify the resulting radicand.

Examples:

1. √3 · √2

Solution:

√3 · √2 = √(3·2) = √6 Product Rule for Radicals

2. √7 · √14

Solution 1:

√7 · √14 = √(7·14) Product Rule for Radicals

= √98 Factor out the greatest square power of 98

= √(49·2)

= 7√2 Simplify

Solution 2:

√7 · √14 = √7 · √(7·2) Factor the radicand

= √(7²·2) Product Rule for Radicals

= 7√2 Simplify

3. ⁴√(4y³) · ⁴√(12y²)

Solution:

⁴√(4y³) · ⁴√(12y²) = ⁴√(48y⁵) Product Rule for Radicals

= ⁴√(16y⁴ · 3y) Factor out the greatest fourth power

= 2y ⁴√(3y) Simplify

4. 4∛3(2∛6 − 7∛5)

Solution:

4∛3(2∛6 − 7∛5) = 4∛3 · 2∛6 − 4∛3 · 7∛5 Distributive Property

= 8∛18 − 28∛15 Product Rule for Radicals

5. (√5 − 2√15)(√5 + √15)

Solution:

(√5 − 2√15)(√5 + √15) = √5(√5 + √15) − 2√15(√5 + √15) Distributive Property

= √25 + √75 − 2√75 − 2√225 Product Rule for Radicals

= 5 + 5√3 − 10√3 − 30 Simplify

= −25 − 5√3 Combine like terms

6. Compare the volume of the cubes.

[illustration: two cubes labeled A and B; the side of cube A is labeled √x and the side of cube B is labeled ⁶√x³]

Solution: The formula to find the volume of a cube is V = s³.

s_A = √x s_B = ⁶√x³

V_A = (√x)³ V_B = (⁶√x³)³ (substitute the value of s to the formula V = s³)

V_A = √x³ V_B = ⁶√x⁹

V_A = x√x V_B = √x³

V_B = x√x

The two cubes have the same volume.

When the indices are different,

Do the following:

1. Write the radicals as expressions with rational or fractional exponents.
2. Change the fractional exponents into similar fractions.
3. Rewrite the product as a single radical.
4. Simplify the resulting radicand if necessary.

Examples:

1. ∛3 · √3

Solution:

∛3 · √3 = 3^(1/3) · 3^(1/2) Write using rational exponents

= 3^(2/6) · 3^(3/6) Write the exponents in similar fractions

= ⁶√(3² · 3³) Write as a single radical

= ⁶√3⁵ Product of Powers Rule

= ⁶√243 Simplify

2. √2 · ∛2 · ⁴√2

Solution:

√2 · ∛2 · ⁴√2 = 2^(1/2) · 2^(1/3) · 2^(1/4) Write using rational exponents

= 2^(6/12) · 2^(4/12) · 2^(3/12) Write the exponents as similar fractions

= ¹²√(2⁶ · 2⁴ · 2³) Write as a single radical

= ¹²√(2¹³) Product of powers rule

= ¹²√(2¹² · 2) Factor out the greatest 12th power

= 2 ¹²√2 Simplify

3. √2 · ∜4

Solution:

√2 · ∜4 = 2^(1/2) · 4^(1/3) Write using rational exponents

= 2^(3/6) · 4^(2/6) Write the exponents in similar fractions

= ⁶√(2³ · 4²) Write as a single radical

= ⁶√(2³ · (2²)²) 4 can be expressed in base 2, 4 = 2²

= ⁶√(2³ · 2⁴) Power of a Power Rule of Exponents

= ⁶√2⁷ Add the exponents

= ⁶√(2⁶ · 2) Extract the radicand with power of 6

= 2 ⁶√2 Simplify

4. 5∛(a²) · 7⁵√(a⁶)

Solution:

5∛(a²) · 7⁵√(a⁶) = 5(a)^(2/3) · 7(a)^(6/5) Write using rational exponents

= 5(a)^(10/15) · 7(a)^(18/15) Write the exponents as similar fractions

= 5 · 7 · ¹⁵√(a¹⁰ · a¹⁸) Write as a single radical

= 35 ¹⁵√(a²⁸) Add the exponents

= 35 ¹⁵√(a¹⁵ · a¹³) Factor out the greatest 15th power

= 35a ¹⁵√(a¹³) Simplify

Division of Radicals

Rule:

To divide a radical by another radical, rewrite the expression into fractional form and then rationalize the result like what we do in simplifying radicals involving fractions. In some cases, it is much easier to apply the Quotient Rule for Radicals, that is ⁿ√a / ⁿ√b = ⁿ√(a/b), before rationalizing the denominator. To rationalize a denominator that has two terms where one or all of them having a square root or radical, we multiply the denominator by its conjugate. Conjugates are the sum and difference of the same two terms. The product of two conjugates results in a difference of two squares and has no radicals.

Examples:

1. √72 ÷ √2

Solution 1:

√72 ÷ √2 = √72/√2 Rewrite in fraction form

= (√72/√2) · (√2/√2) Rationalize the denominator by multiplying √2/√2 since √2 · √2 = 2

= √144/√4 Multiply

= 12/2 Simplify the radicals

= 6

Solution 2:

√72 ÷ √2 = √(72/2) Quotient Rule for Radicals

= √36 Simplify

= 6

2. √6x ÷ √3x

Solution:

√6x ÷ √3x = √(6x/3x) Quotient Rule for Radicals

= √2 Simplify

3. ∛(18y²) ÷ ∛(12y)

Solution:

∛(18y²) ÷ ∛(12y) = ∛(18y²/12y) Quotient Rule for Radicals

= ∛(3y/2)

= (∛(3y) · ∛(2²)) / (∛2 · ∛(2²)) Rationalize the denominator by multiplying ∛(2²)/∛(2²) since ∛2 · ∛2² = 2

= ∛(12y)/2 Simplify

4. √2 ÷ (2 + √3)

Solution:

√2 ÷ (2 + √3) = √2/(2 + √3) Rewrite in fraction form

= (√2/(2 + √3)) · ((2 − √3)/(2 − √3)) Rationalize the denominator by multiplying to its conjugates

Take note that conjugates are the sum and difference of the same two terms. The product of two conjugates results in a difference of two squares and has no radicals.

= (2√2 − √6)/(4 − 3) Distributive Property

= (2√2 − √6)/1 Simplify

= 2√2 − √6 Write in lowest term

5. √(xy) ÷ (√x − √y)

Solution:

√(xy) ÷ (√x − √y) = √(xy)/(√x − √y) Rewrite in fraction form

= (√(xy)/(√x − √y)) · ((√x + √y)/(√x + √y)) Rationalize the denominator by multiplying to its conjugates

= (√(x²y) + √(xy²))/(x − y) Distributive Property

= (x√y + y√x)/(x − y) Simplify

6. The area A of a parallelogram is 7√12 square units while its base b is 7√3 units. Find the height of the parallelogram.

Solution: The area of a parallelogram is given by the formula A = bh. To find the height of the parallelogram, we need to divide its area by the measure of the base.

h = A ÷ b

h = 7√12 ÷ 7√3 = 7√12/7√3 = √12/√3 = √(12/3) = √4 = 2 units

The height of the parallelogram is 2 units.

What is It

Perform the operation on radical expressions with the problem solving maps as guide.

[chart: multi-rule map (MRM) worksheet; ACTIVITY 1 Addition and Subtraction of Radical Expressions]

ACTIVITY 1 ADDITION AND SUBTRACTION OF RADICAL EXPRESSIONS STUDENT’S WORKSHEET ON MULTI-RULE MAP (MRM)

Starting box: 3√20 − 2√24 + √180 − √54

Five stacked empty answer boxes run down the left side, with four empty cloud-shaped boxes beside them.

[transcriber’s note: the answer boxes and cloud shapes are blank in the source]

What’s More

[chart: math-breaker map (MBM-structured) worksheet; ACTIVITY 2 Multiplying Radical Expressions]

ACTIVITY 2 MULTIPLYING RADICAL EXPRESSIONS STUDENT’S WORKSHEET ON MATH-BREAKER MAP (MBM-STRUCTURED)

Starting box: 2√5 · √7

Step boxes:

• Multiply the coefficients of each radical expression
• Multiply the radicands of each radical expression
• Combine the product of the coefficients and the product of the radicands
• Simplify

[transcriber’s note: the answer boxes below each step are blank in the source]

What I Have Learned

Operations with radicals include:

1. Addition and Subtraction: Combine like radicals by adding or subtracting their coefficients and copy their common radicand. The sum or difference of unlike radicals, unless written to equivalent or like radicals, cannot be expressed as a single term.
2. Multiplication: Multiply the coefficients and multiply the radicands then simplify. The product rule for radicals is applied for multiplying radicals.
3. Division: For all radicals whose denominators are not equal to zero, divide the coefficients and divide the radicands then simplify. The quotient rule for radicals is applied when dividing radicals.
4. To rationalize a denominator that has two terms where one or all of them having a square root or radical, we multiply the denominator by its conjugate. Conjugates are the sum and difference of the same two terms. The product of two conjugates results in a difference of two squares and has no radicals.

What I Can Do

[chart: math-breaker map (MBM-structured) worksheet; ACTIVITY 3 Dividing Radicals]

ACTIVITY 3 DIVIDING RADICALS STUDENT’S WORKSHEET ON MATH-BREAKER MAP (MBM- STRUCTURED)

Starting box: ∛10 ÷ ∛2

Step boxes, in order: Write in fraction form → Rationalize the denominator → Multiply → Apply Quotient Rule for Radicals → Simplify.

[transcriber’s note: the answer boxes below each step are blank in the source]

Assessment

Read the questions carefully. Write the letter of the correct answer.

1. When do you say that two radicals are alike?
   a. when their indices are the same
   b. when their radicands are the same
   c. when their integral exponents are the same
   d. when their indices and radicands are the same

   
   

2. The product of (5√2 + 4)(5√2 − 4) is __________.
   a. -34 b. 34 c. 26 d. -26

   
   

3. Which of the following set of radicals can be added or subtracted?
   a. √(xy), 3√(5x) c. 3m√n, m³√n
   b. 2√(7x), 3√(63x) d. ∛(xy²), 3∛(x²y)

   
   

4. Three of the radicals below can be expressed as similar radicals. Which of the following is NOT similar to the other given radicals?
   a. √44 b. √99 c. √110 d. √176

   
   

5. The combined form of 64√3 − 4√48 − 5√48 is ____.
   a. 3√8 b. 28√3 c. 8√3 d. 3√48

   
   

6. Divide: ∛(48x³y⁷) / ∛(8x⁵y)
   a. (x³∛8)/y b. (y²∛6x)/x c. (x²∛6)/y d. (y³∛6)/x

   
   

7. What is the equivalent of ∛4 + ⁵√2 using exponential notation?
   a. 4^(1/3) + 2^(1/5) b. 4³ + 2⁵ c. 6⁸ d. 6^(1/8)

   
   

8. What is the result after simplifying 2√3 + 4√3 − 5√3?
   a. −√3 b. √3 c. 11√3 d. 21√3

   
   

9. Evaluate (2⁻⁴)(2¹⁰).
   a. 2¹⁴ b. 2⁻¹⁴ c. 64 d. 1/2

   
   

10. The length of a rectangle is 2√50 meters and its width is 3√8 meters. Find its area.
    a. 5√58 sq. meters c. 32√2 sq. meters
    b. 120 sq. meters d. 124 sq. meters

    
    

Additional Activities

What’s the Message?

Decode the message by performing the operations on radicals. Write the letter that corresponds to the answer in the box provided for.

√7 + √7	2√6 − 4√6	5∛3a + 5∛3a − 9∛3a	6√2 − 5√8 + 2√32
A	B	C	D

5√27 − 2∛27 + ∛27	√12 − 5∛8 − 7√20	√4 · √6	5∜a · 4∜a³
E	I	L	R

∜3(∜2 + ∜4)	√75 ÷ √3	∛4x² ÷ ∛x	1 ÷ (√x − √y)
S	T	U	Y

Answer Box

2√7	⁴√3a [?]	⁴√3a [?]	∛4x	20a⁴√a⁵ [?]	2√7	⁴√3a [?]	(√x + √y)/(x − y)

−2√6	∜4x	2√3 − 10 − 14√5	2√6	4√2	⁴√6 + ⁴√12

⁴√3a [?]	20a⁴√a⁵ [?]	15√3 − 3	4√2	2√3 − 10 − 14√5	−2√6	2√3 − 10 − 14√5	2√6	2√3 − 10 − 14√5	5	(√x + √y)/(x − y)

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=4Gq3LPORQU https://www.youtube.com/watch?v=NxRDUc3p0QQ

Prentice Hall Algebra 1 Teaching Resource https://1.cdn.edl.io/x1HxFjyVLo8pgAr2evQRFJb68knnbUOXdiUn0W67oRsODBSt.pdf http://jeff560.tripod.com/mathword.html

PROBLEM – BASED WORKSHEET

TLE PROJECT

Norman and Benedict are Grade 9 students. Mr. Rivera, their TLE teacher, asked them to bring wood boards of any shapes and sizes to be used in making utility boxes. Norman brought a triangular board while Benedict brought a rectangular board whose dimensions are shown below.

√48 dm √27 dm √125 dm √48 dm

LET’S ANALYZE

1. What grade level are Norman and Benedict in?
2. What is their project in TLE?
3. What shape did Norman use for the base of his box? What about Benedict?
4. If the boxes should be 5 dm in height, how much wood would be needed to make the sides of each box?
5. Whose utility box has the more space? Why did you say so?

Answer Key

ACTIVITY 1

ADDITION AND SUBTRACTION OF RADICAL EXPRESSIONS STUDENT’S WORKSHEET ON MULTI-RULE MAP (MRM)

[chart: flowchart; ACTIVITY 1, Addition and Subtraction of Radical Expressions, Student’s Worksheet on Multi-Rule Map (MRM)] Steps, from start to end:

1. Factor the radicands.
2. Get the roots of each factor.
3. Group similar radicals.
4. Add the coefficients of similar radicals.

Boxes, from start to end:

• 3√20 − 2√24 + √180 − √54
• 3√(4·5) − 2√(4·6) + √(36·5) − √(9·6)
• 6√5 − 4√6 + 6√5 − 3√6
• (6√5 + 6√5) − (4√6 + 3√6)
• 12√5 − 7√6

WHAT I KNOW

1. B
2. C
3. C
4. C
5. B
6. B
7. B
8. C
9. B
10. A
11. D
12. D
13. A
14. C
15. B

WHAT’S IN

1. 7
2. 3
3. −4n
4. 5x^(3/4)/√2 [?]
5. [illegible]
6. [illegible]
7. ∛√8 [?]
8. −2y/x⁴ [?]
9. 3√3 [?]
10. ∛(√54)/(5a²) [?]

ASSESSMENT

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

ADDITIONAL ACTIVITY

10. B

“Accuracy Builds Credibility”

PISA – BASED WORKSHEET

1. Grade 9
2. Making a utility box
3. Norman used a triangular base while Benedict’s box has a rectangular base.
4. Norman’s box needed 20√3 + 25√5 + 5√175 sq. dm. of wood while Benedict’s box needed 70√3 sq. dm. of wood.
5. Norman’s box has more space because it has a bigger volume.

WHAT I CAN DO

ACTIVITY 3

DIVIDING RADICALS STUDENT’S WORKSHEET ON MATH-BREAKER MAP (MBM- STRUCTURED)

[chart: flowchart; Activity 3, Dividing Radicals, Student’s Worksheet on Math-Breaker Map (MBM-Structured)] Steps, from start to end:

• ∛10 ÷ ∛2
• Write in fraction form: ∛10/∛2
• Rationalize the denominator: (∛10/∛2) · (∛4/∛4)
• Multiply: ∛40/∛8
• Apply Quotient Rule for Radicals: ∛(40/8)
• Simplify: ∛5

WHAT’S MORE

ACTIVITY 2

MULTIPLYING RADICAL EXPRESSIONS STUDENT’S WORKSHEET ON MATH-BREAKER MAP (MBM-STRUCTURED)

[chart: flowchart; Activity 2, Multiplying Radical Expressions, Student’s Worksheet on Math-Breaker Map (MBM-Structured)] Steps, from start to end:

• Start: 2√5 · √7
• Branch 1: Multiply the radicands of each radical expression: √5 · √7 = √35
• Branch 2: Multiply the coefficients of each radical expression: 2 · 1 = 2
• Combine the product of the coefficient and the product of the radicands: 2 · √35
• Simplify: 2√35

References

Nivera, G.C. & Lapinid, M. C. (2013). Grade 9 Mathematics Patterns and Practicalities. Salesiana Books by Don Bosco Press 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) Validated Problem Solving Maps Worksheet

For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resources (DepEd-BLR)

Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph