(Effective Alternative Secondary Education)

MATHEMATICS II

MODULE 5

Radical Expressions

BUREAU OF SECONDARY EDUCATIONDepartment of Education

DepEd Complex, Meralco Avenue, Pasig City

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Module 5Radical Expressions

What this module is about

Just as you did in the case of adding and subtracting radical expressions, this module will allow you to multiply and divide them by applying the same basic procedures in dealing with algebraic expressions. You will constantly be using properties of radicals which is in the box for easy reference.

What you are expected to learn

1. Recognize basic radical notation2. apply the basic properties of radicals to obtain an expression in

simplest radical form.3. multiply and divide radical expressions.

How much do you know

A. Multiply the following expressions.. _ _

1. 43 . 33 _ _

2. 57 . 27 _ _

3. 25. 7 _ _

4. 52 . 5 __ _

5. (2x2b)(5b )

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__ _ _Property 1 ab = a . b _Property 2. a = a b b

B. Divide the following expressions. _ _

1. 2 3 _ _

2. 34 36_ _

3. 2 32 _ _

4. 2 (2 + 3) __ _ _

5. xy (x - y)

What will you do

Lesson 1

Multiplication of Radical Expression

In multiplying radical, there are three cases to be considered. These are:

a. Indices are the same. When multiplying radicals having the same index,

_ _ __apply: nx . n y = nxy and then if necessary, simplify the resulting radicand.

b. Indices are different but radicands are the same. To find the product of radicals with different indices, but the same radicand, apply the following steps:

1. transform the radical to fractional exponents.2. multiply the powers by applying: xm . xn = xm+n (law of exponent)3. rewrite the product as a single radical.4. simplify the resulting radicand if necessary.

c. Indices and radicands are different. To find the product with different indices and radicands, follow the following steps:

1. transform the radicals to powers with fractional exponents.2. change the fractional exponents into similar fractions.3. rewrite the product as a single radical4. Simplify the resulting radicand if necessary.

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Multiplying monomial radicals

Rules to follow:Rule 1. If radicals to be multiplied have the same indices, follow the steps

in the examples. _ _ _Example 1. Multiply: 2.3.5

Solution: Write the product of two or more radicals as a single expression. _ _ _ ____

2.3.5 = 2.3.5 __ = 30 __ __

Example 2. Find the product: 12 . 18Solution: There are two approaches to solve.

__ __ _____12 . 18 = 12.18 by property 1

___= 216 Look for the largest perfect square

factor of 216, which is 36. __ _

= 36 . 6 _= 66

Second approach: First put each radical into simplest form. __ __ _ _ _ _ 12 . 18 = 4. 3 . 9. 2

_ _= 23 . 32 Rearrange the factors. _ _= 2.33 2 _= 6 6

Note that the second approach used kept numbers much smaller. The arithmetic was easier when the radical is simplified first.

_ __Example 3. Find the product: 7 . 14

Solution: _ __ ____7 . 14 = 7.14

__ = 98 express the radicand as product

of the largest perfect square factor.

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__ _= 49. 2 _= 72

_ _Example 4. Multiply: a3 . b6

Solution: _ _ ___a3 . b6 = ab3.6 simply multiply the radicand

having the same index. __

= ab18 express the radicand as product of the largest square factor

_ _ = ab9 . 2

_ = 3ab2 ___ ____

Example 5. Get the product: 2ab3 . 12abSolution: ____ ____ ___________

2ab3 . 12ab = (2ab3).(12ab) applying the law of exponent

_____ = 24a2b4 expressing the radicand

as the largest square factors

_ _ _ _ = 4 .6 a2 b4

_ = 2ab2 6

Rule 2. If the radicals have different indices but same radicands, transform the radicals to powers with fractional exponents, multiply the powers by applying the multiplication law in exponents and then rewrite the product as single radical.

_ _Example 6. 5 . 4 5 _ _

Solution: 5 . 45 = 51/2 . 51/4

= 5 ½ + ¼

= 53/4

__ ___ = 453 or 4125

____ ____

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Example 7. (42x – 1) ( 32x – 1Solution:

_____ _____ (42x – 1 ) ( 32x – 1) = (2x -1 )1/4 (2x – 1)1/3

= (2x – 1) ¼ + 1/3

= (2x – 1) 7/12

_______ = 12(2x – 1)7

Rule 3: If radicals have different indices and different radicands, convert the radicals into powers having similar fraction for exponents and rewrite the product as a single radical. Simplify the answer if possible.

_ _Example 8. 2 33

Solution: _ _ 2 33 = 21/2 . 31/3

= 23/6 . 32/6

__ __ = 623 . 632

____ = 6 8 . 9

__ = 672

_ _Example 9. 42 . 35

Solution: _ _ 42 . 35 = 21/4 . 51/3

= 23/12 . 54/12

__ __ = 1223 . 1254

_ ___ = 128 . 12625

_____ = 12 5000

Multiplying a radical by a binomial

In each of the following multiplication, you are to use the distributive property to expand the binomial terms.

_ _ _Example 10. Multiply: 3 ( 23 + 5)

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Solution: Using the distributive law, then _ _ _ _ _ _ _

3 (23 + 5) = 3 . 23 + 3 . 5 _ _ ____= 23.3 + 3. 5

___= 2.3 + 15

__= 6 + 15

_ _ _Example 11. Multiply and simplify: 2x (x - 3) – 4(3 - 5x)

Solution: Proceed as if there are no radicals- using the distributive law to remove the parentheses;

_ _ _ _ _ _ _ 2x (x - 3) – 4(3 - 5x) = 2x x - 6x – 12 + 20x

_ _ = 2 x -6x – 12 + 20x

_ _ = 2x - 6x–12 + 20x combine like terms

_ = 2x + 14x – 12

Binomial Multiplication.

This method is very much similar to the FOIL method. The terms are expanded by multiplying each term in the first binomial by each term in the second binomial. _ _ _ _Example 12. (43 + 2) (3 -52

_ _ _ _Solution: (43 + 2) (3 -52)

Use the FOIL method, that is multiplying the first terms, outer terms, inner terms and the last terms. _ _ _ _ _ _ _ _

= 4(3)(3) -43(52) + 2(3) - 2(52) _ _ _ _= 4(3)2 - 206 + 6 -5(2)2

_ _= 4 . 3 - 206 + 6 – 5 . 2 _ _= 12 - 206 + 6 – 10

_= 2 - 196

_ _ _ _Example 13. (a + 3) (b + 3)

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_ _ _ _ Solution: (a + 3) (b + 3) FOIL these binomial then

simplify. _ _ _ _ _ _ _ = ab + 3a + 3b + (3)2

__ __ __ = ab + 3a + 3b + 3

_ _Example 14. Multiply and simplify: (7 - 3 )2

Solution. Watch out! Avoid the temptation to square them separately.

_ _ _ _ _ _(7 - 3)2 = (7 - 3) (7 - 3)

_ _ _ _ _ _ _ _ = 7 7 - 7 3 - 7 3 + 3 3

__ __ = 7 - 21 - 21 + 3 Combine like terms

__ = 10 - 221 _ ____

Example 15. (a – 3)2 – (a – 3 )2

Solution: Note the difference between the two expressions being squared.

The first is a binomial; the second is not. _ ___ _ _ ____ ___

(a – 3)2 – (a – 3 )2 = (a – 3)(a – 3) - a – 3 a-3 _ _ _ _

= aa - 3a -3a + 9 – (a - 3)

Note that the parentheses around a – 3 is essential. _

= a – 6 a + 9 – a + 3 _

= -6a + 12

Multiplying Conjugate Binomials

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Remember: (a+b)2 a2+ b2

The product of conjugates are always rational numbers. The product of a pair of conjugates is always a difference of two squares (a2 – b2), multiplication of a radical expression by its conjugate results in an expression that is free of radicals.

__ __Example 16. (13 -3) (13 + 3)

Solution: Multiply out using FOIL. __ __ __ __ __ __

(13 -3) (13 + 3) = 13 13 + 313 - 313 – 9 The middle terms combine to 0. = 13 – 9

= 4 This answer does not involve radical.

_ _ _ _ Example 17. (5 + 7 ) (5 - 7) A difference of squares

_ _ A square of a root is the= (5)2 – (7)2 original integer

= -2 Simplified _ _ _ _

Example 18. (7 + 23)(7 - 23) _ _= (7 )2 – (23 )2

= 7 – 12

= -5

Try this out

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Perform the indicated operations. Simplify all answers as completely as possible.

A. _ __ _ __ 1. 311 6. 5 45 _ _ __ _ _ __2. 3513 7. 2610 _ _ _ _ _3. 6 24 8. 3 5 6 __ __ __ __4. 18 32 9. 24 28

_ _5. (-42 )2 10. ( 35 )2

B. __ _ _ _ _ _ 11. 25c . 55 16. (23 - 7)(23 + 7)

_ _ _ ____12. 25 (53 + 35) 17. ( 1 + x + 2 )2

_ _ _ _ _13. (25 -4)( 25 + 4) 18. 3 ( 23 - 32)

_ _ _ _ _ _ _ _14. (33 - 2) ( 2 + 3) 19. 32 (2 – 4)+ 2 (5 - 2)

_ _ _ _15. (3 + 2) (3 -5) 20. (x + 3 )2

C. What’s Message?

Do you feel down with people around you? Don’t feel low. Decode the message by performing the following radical operations. Write the words corresponding to the obtained value in the box provided for.

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are not _ _2 . 58

and irreplaceable _ _37 . 47

consider yourself _ _43 . 33

Do not _ _9 . 4

Each one ____ _____39xy2 . 3 33x4y6

for people (43a3)2

is unique _ __3 . 318

more or less __ _27 . 3

nor even equal _ _a (a3 – 7)

of identical quality _ _57 . 27

to others __ __ ___(5a)(2a)(310a2)

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2048a3

36 9 _a2-7a

30a2

_127

___9xy2 3x2y2

___654

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Lesson 2

Division of radicals

Dividing a radical by another radical, follows the rule similar to multiplication. When a rational expression contains a radical in its denominator, you often want to find an equivalent expression that does not have a radical in the denominator. This is rationalization. Study the following examples.

__Example 1. Simplify: 72 6

Solution: You are given two solutions: __

b. Simplify 72. b. Make one radical expression ___ __ __ __ 72 = 36 2 72 = 72 6 6 6 6

_ __ = 6 2 Rationalize = 12

6 _ _ _ _ = 4 3

= 6 2 . 6 _ 6 6 = 23

__ = 6 12

6 __

= 12_ _

= 4 3 _

= 23

Note: Clearly the second method is more efficient. If you have the quotient of two radical expressions and see that there are common factors which can be reduced, it is usually method 2 is a better strategy, first to make a single radical and reduce the fraction within the radical sign. then proceed to simplify the remaining expression. ___Example 2. 6b 7 _ 30ab

___ Solution: 6b 7 __ = 6b 7 Reduce

30ab 30ab

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b 6 = 5a

_ = b 6 5a

= b 3 5a

__ = b 3 . 5a 5a 5a

__ = b 3 5a

5a

Rationalizing binomial denominators

The principle used to remove such radicals is the familiar factoring equation. If a or b is square root, and the denominator is a + b, multiply the numerator and the denominator by a – b and if a or b is a square root and the denominator is a + b, multiply the numerator and the denominator by a – b. (a + b) (a – b) = a2- b2

Example 3. ___2___ 7 - 5 __ __

Solution: the denominator is 7 - 5, is the difference, so multiply the numerator and the numerator by the sum 7 + 5:

_ _ _ _ ___2___ x 7 + 5 = 2( 7 + 5)

7 - 5 7 + 5 (7)2 – (5)2

_ _ = 2( 7 + 5 ) _

7 – 5 _ _

= 2( 7 + 5) Simplify 2

_ _ = 7 + 5

Example 4. ___20___ 10 + 6

__ _Solution: ___20___ = ___20___ . 10 - 6 10 + 6 10 + 6 10 - 6

__ _ = 20( 10 - 6)

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10 – 6 __ _

= 20( 10 - 6 ) 4 __ __ _ _= 5(10 - 6) or 510 - 56

Example 5. Simplify as completely as possible: ___8___ - 10 3 - 5 5

Solution: Begin by rationalizing each denominator. Keep in mind that each fraction has sits own rationalizing factor.

_ ____8___ - 10 = ___8___ . 3 + 5 - 10 . 5

3 - 5 5 3 - 5 3 + 5 5 . 5 _ _

= 8(3 + 5 ) - 10 5 Reduce each fraction 9 – 5 5

_ _= 8(3 + 5 ) - 10 5 Simplify the numerator

4 5 and denominator which _ _ are not radicand.= 2(3 + 5) - 25 Combine similar radicands. _ _= 6 + 25 - 25= 6 __

Example 6. Simplify: 12 + 18 6

Solution: Begin by simplifying the radical. __ _ _

12 + 18 = 12 + 9 2 6 6

_ = 12 + 3 2 Factor out the common factor

6 of 3 in the numerator. _

= 3(4 + 2) simplify 6

_ _ = 4 + 2 or 2 + 2

2 2 _ _

Example 7. 2 32 _ _ _

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Solution: 2 32 = __ 2__ 32

= 2 1/2 Change the radicals to fractional exponent. 21/3

= 2 3/6 Change the fractional exponents to similar 22/6 fractions

= 6 2 3 Transform the expression as a single radical. 22 and simplify. _ = 62

_____ _______Example 8. Express as a single radical: 4xy2z2 616xy2z4

_____ _______ Solution: 4xy2z2 616xy2z4 Transform to fraction

_____= __ 4xy 2 z 2 __ 616xy2z4

= (4xy 2 z 3 ) 1/2 Change to fractional exponent (16xy2z4)1/6

= (4xy 2 z 3 ) 3/6 Change the fractional (16xy2z4)1/6 exponent to similar fractions. _______

= 6 (4xy 2 z 2 ) 3 Rewrite as radical expressions 616xy2z4 the radicand to powers.

= 6 64x 3 y 6 z 6 Simplify. 16xy2z4

= 6 4x2y4z2

_ _Example 9. Perform: 2 (2 + 3)

_ _ _Solution: 2 (2 + 3) = __ 2__ rewrite the expression

2 + 3

_ _ = __ 2 __ . 2 - 3 rationalize

2 + 3 2 - 3

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_ _ = 2 2 - 6 simplify 4 - 3

_ _ = 22 - 6 __ _ _

Example 10. Simplify: xy (x - y) __ _ _ __

Solution: xy (x - y) = __ xy __ rewrite the expression(x - y) __ _ _

= __ xy __ . x + y rationalize x - y x + y

___ ___ = x 2 y + xy 2

x – y _ _

= x y + y x x – y

Try this out

A. Divide and simplify __ __ 1. 618 1240

__ __ 2. 819 438

_ _3. 203 53

_ _4. 426 36

__ _5. -420 2

_ _6. 1018 29

__ __ 7. 596 224 __ __ 8. 3/7 30 1/3 15

__ __9. 2046 523

_ __10. 63 18 _ __11. 122 227

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_ __12. 126 ¼ 72 __ ___13. 50 125 __ ___14. 45 400

15. 3 3x2b 4 25xy2

B. Simplify __

1. 10 3 2 _

2. 3 3 3

3. 4 3 3 3 4. 3

6 4 6

5. 3 36 4 6 _ _

6. 9 3

7. 4 27 3 2

8. __1__ 2 + 59. __1__ 3 - 11

10. __3__ 3 – 1

D. Why is tennis a noisy game? Solve the radicals by performing the indicated operation. Find the answer below and exchange it for each radical letter.

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_ _ 2 3

_ _ 38 36

__ _ 436 46

_ _ 34 36

___7___ 6 + 5

_ __ 4 6 - 3 21 3

_ _ 2 32

_ __ 5 15

____ ____ 3 3x2b 325xy2

_ _ 22 (2+3)

_1_ x

___ _ 3 108 32

__ __ 563 67

400 20

_1_ 5

__ 6 28 34

__ _ 80 5

__ 20 46 523

__1__ 2+5

_ _ 1018 29

__ __ 5 96 2 24 6 5

__3__ 3 - 1

__ ___ 25 625

_ _ 33 35

6 3

x 2 7 4

_52

3 36 3 62 5

2

_______

3 15bxy 5y

_ 5 5

_ 5 5

_4 6

3 3+3 2

3 18 3 2

_27

3 75 5

_42

_-2+5

_25

3 3

_ _47-37

_3 32

25 6

Let us summarize

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Definition:

The pairs of expressions like x - y and x + y or x - y and x + y are called conjugates. The product of a pair of conjugates has no radicals in it. Hence, when we rationalize a denominator that has two terms where one or more of them involve a square-root radical, we multiply by an expression equal 1, that is, by using the conjugate of the denominator.

What have you learned

A. Fill in the blanks.

1. For a = b2, _______is the square root of ______.2. When no index is indicated in a radical, then it is understood that the index

is _____.3. In radical form, 169 3/2 is written as ____ or ____

__4. In simplest form. 54 is ____ __5. In simplest form 316 is ____ __6. In simplest form 464 is ____ __7. in simplest form, 616 is ___ _____8. In simplest form 50x7y11

____ ____ 9. The product of (3 2 + 4)(32 – 4)

__ __10.The product 26 . 44 _ __ __11.The combined form 57 -228 - 348 is ___________.

__12. In simplest form, the quotient 27 = _______

48 ___

13. In simplest , the quotient 3135 = _____ 340 _14. In simplest form, the quotient __ 7 __

3 - 2 ______ 15. In simplest form, the quotient 4 162x 6 y 7 = ____

432x8y

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Answer Key

How much do you know:

A. 1. 3.4.3 = 362. 5.2.7 = 70 __3. 235

4.5105. 10bx _

B. 1. 6/3 __

2. 318/3 _3. 62 _ _4. 22 - 6 _ _

5. x y + y x x-y

Try this out

Lesson 1 __A. 1. 33 ___ 2. 199

3. 12 __

4. 1212

5. 32

6. 15 _

7. 23 __

8. 310 __

9. 242

10.45

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_B. 11. 50c __

12. 1015 + 30

13. 4 _ _ 14. 26 + 7 or 7 + 26 _ _

15. -33 – 7 or -7 - 33

16. 5 ____

17. 3 + x + 2 x + 2 _ 18. 6 - 36 _

19. 4 - 72 __

20. 3 + x + 23x

C.

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Do not consideryourself

more or less nor even equal

to others

for people are not of identical quality

each one Is unique

and are irreplaceable

Lesson 2

Try this out. _ _A. 1. 3 5 11. 2 6 20 3 __ __ 2. 2 19 12. 12 19 4

__ 3. 4 13. 10

5_

4. 14 14. 3 5

__ _____ 5. - 410 15. 4 75bxy 2 _ 5y 6. 5 2

7. 10 _

8. 9 2 2

_9. 42

_10. 6

___ _B. 1. 6250 8. -2 +5 ___ __ 2. 6 243 9. 3 + 11 3 -2 __ _ 3. 12 3 11 10. 1 + 3 3 3 2 _ 4. 126 __ ____ 5. 1265 or 127776 _ 6. 3 ___ 7. 12243

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C. Why is tennis a noisy game?

6 3

x 2 7

4

_52

3 36 3

_62

5 6

3 15xby 5y

5 5

5 5

E V E R Y P L A Y E R

R A I S E S A R A C K E T _46

3 3+3 2

3 18 3 2 27

3 75 5

_ 42 -2+5

_25

3 3

_ _42-37

_3 32

25 6

What have you learned

A. 1. a,b2. 2 ____ ________3. 169 3 or 4826809 _4. 36 _5. 2 3 2

_6. 2 4 4

__7. 664

___8. 5x3y5 2xy

9. 18 + 9x

10. 6 16 _ _

11. 7 - 123

12. 3/4

13. 3/2 __ __

14. 21 + 14 ______15. 3 4 72x 2 y 6

4x

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