MATH 2 part 2

7. 12√243C. Why is tennis a noisy game?√6 √x _ 3√36 _ 5 3√15xby √5 √5 5√2 3 6√2 6 5y 5 532 74 A YPL YE REVERRA I SE S A R AC K ET _ _ _ 25_ 3√3+3 3√18 3√75 _ _ √3 4√2-3√7 3 3√2 64√6 2 3 2 2√7 5 4√2 -2+√5 2√5 3What have you learnedA. 1. a,b 2. 2 ____ ________ 3. √169 3 or √ 4826809 _ 4. 3√6 _ 5. 2 3√ 2 _ 6. 2 4 √4 __ 7. 6√64 ___ 8. 5x3y5 √2xy 9. 18 + 9x 10. 6 √16 __ 11. √7 - 12√3 12. 3/4 13. 3/2 __ __ 14. √21 + √14 ______ 15. 34√72x2y6 4x 23

Module 5 Radical ExpressionsWhat 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 basicprocedures in dealing with algebraic expressions. You will constantly be usingproperties of radicals for easy reference.Property 1: ab = a · bProperty 2: a= a bbWhat you are expected to learn1. 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 knowA. Multiply the following expressions.. 1. 4 3 · 3 3 2. 5 7 · 2 7 3. 2 5 · 7 4. 5 2 · 5

5. (2 x2b ) (5 b ) B. Divide the following expressions. 1. 2 ÷ 3 2. 3 4 ÷ 3 6 3. 2 ÷ 3 2 4. 2 ÷ (2 + 3 5. xy ÷ ( x - y ) What will you do Lesson 1 Multiplication of Radical Expression In multiplying radicals, there are three cases to be considered. These are:a. when indices are the same. Multiply radicals and apply n x ⋅ n y = n xy Simplify the resulting radicand when necessary.b. when indices are different but radicands are the same. Apply the following steps: 1. transform the radical to fractional exponents. 2. multiply the powers by applying: xm · xn = xm+n (law of exponents) 3. rewrite the product as a single radical. 4. simplify the resulting radicand if necessary.c. when indices and radicands are different. Do 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 radical 4. Simplify the resulting radicand if necessary. 2

Multiplying Monomial Radicals Rules to follow: Rule 1. For radicals having the same indices, follow the steps in the examples.Examples:Find the product of the following:1. 2 · 3 · 5 Solution: Write the product of two or more radicals as a single expression.2 · 3 · 5 = 2⋅3⋅5= 302. 12 · 18 Solution: There are two approaches to solve. First Approach:12 · 18 = 12 ⋅18= 216 Find the largest perfect square factor of 216, which is 36.= 36 · 6=6 6Second approach: First put each radical into simplest form.12 · 18 = 4 · 3 · 9 · 2=2 3 ·3 2 Rearrange the factors.=6 6 3

3. 7 · 14 Solution:7 · 14 = 7 ⋅14 = 98 Express the radicand as product of the largest perfect square factor. = 49 · 2 =7 24. a 3 · b 6 Solution:a 3 · b 6 = ab 3 ⋅ 6 Multiply the radicand having = ab 18 the same indices. Express the radicand as product of the largest square factor = ab 9 · 2 = 3ab 25. 2ab3 · 12ab Solution:2ab3 · 12ab = 2ab3 ⋅ 12ab Apply the law of exponents = 24a2b4 Express the radicand as the largest square factors = 4 · 6 a2 b4 = 2ab2 6Rule 2. If the radicals have different indices but same radicands, transform the radicals to exponential form. Multiply the powers by applying the law of exponents and rewrite the product as a single radical. 4

6. 5 · 4 5Solution: 11 5 · 45 = 52 ⋅ 54 2+1 = 54 4 3 = 54 = 4 53 or 4 1257. ( 4 2x −1 ) ( 3 2x −1 )Solution: 11( 4 2x −1 ) ( 3 2x −1 ) = (2x −1)4 (2x −1)3 1+1 = (2x − 1)4 3 7 = (2x − 1)12 = 12 (2x − 1)7Rule 3: If radicals have different indices and different radicands, transform the radicals into exponential form. Change exponents as similar fractions and rewrite the product as a single radical. Simplify the answer if possible.8. 2 3 3Solution: 11 2 33 = 22 · 33 32 = 26 · 36 = 6 23 ⋅ 6 32 = 6 8⋅9 = 6 72 5

9. 4 2 ⋅ 3 5 11 Solution: 4 2⋅3 5 = 24 ⋅ 53 34 = 212 ⋅ 512 = 12 23 ⋅ 12 54 = 12 8 ⋅ 12 625 = 12 5000Multiplying a radical by a binomial In each of the following multiplication, you are to use the distributiveproperty to expand the binomial terms.10. 3 ( 2 3 + 5 ) Solution: Using the distributive property, we have 3 (2 3 + 5 ) = 3 · 2 3 + 3 · 5 = 2 9 + 15 = 2 · 3 + 15 = 6 + 1511. 2 x ( x - 3) – 4(3 - 5 x ) Solution: 2 x ( x - 3) – 4(3 - 5 x ) = 2 x2 - 6 x – 12 + 20 x = 2x - 6 x – 12 + 20 x = 2x - 6 x - 12 + 20 x combine like terms = 2x + 14 x – 12 6

Multiplication of Binomials The terms are expanded by multiplying each term in the first binomial byeach term in the second binomial.12. (4 3 + 2 ) ( 3 -5 2 Solution: (4 3 + 2 ( 3 -5 2 ) Use the FOIL method, that is multiplying the first terms, outer terms, innerterms and the last terms.= 4( 3 )( 3 ) -4 3 (5 2 ) + 2 ( 3 ) - 2 (5 2 )= 4 9 - 20 6 + 6 -5 4= 4 · 3 - 20 6 + 6 – 5 · 2= 12 - 20 6 + 6 – 10= 2 - 19 613. ( a + 3 ) ( b + 3 )Solution: ( a + 3 ) ( b + 3 ) Expand using FOIL= ab + 3a + 3b + 9 Extract the square root= ab + 3a + 3b + 3 and simplify14. ( 7 - 3 )2 Solution. Watch out! Avoid the temptation to square them separately.Remember: (a + b)2 ≠ a2+b2( 7 - 3 )2 = ( 7 - 3 ) ( 7 - 3 ) Combine like terms = 49 - 21 - 21 + 9 = 7 - 21 - 21 + 3 = 10 - 2 21 7

15. ( a – 3)2 – ( a − 3 )2Solution: 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 ) = a2 - 3 a -3 a + 9 – (a - 3)Note that the parentheses around a – 3 is essential. =a–6 a +9–a+3 = -6 a + 12Multiplying Conjugate Binomials The product of conjugates are always rational numbers. The product of apair of conjugates is always a difference of two squares (a2 – b2). Multiplicationof a radical expression by its conjugate results in an expression that is free ofradicals.16. ( 13 -3) ( 13 + 3)Solution: Multiply using FOIL.( 13 - 3) ( 13 +3) = 169 +3 13 -3 13 – 9 The middle terms combine equals 0. = 13 – 9 =417. ( 5 + 7 ) ( 5 - 7 ) Difference of two squares= ( 5 )2 – ( 7 )2 Simplify= 5-7= -218. ( 7 + 2 3 )( 7 - 2 3 ) = ( 7 )2 – (2 3 )2 = 7 – 4(3) = 7 - 12 = -5 8

Try this outPerform the indicated operations. Simplify all answers as completely as possible.A.1. 3 11 6. 5 452. 3 5 13 7. 2 6 103. 6 24 8. 3 5 64. 18 32 9. 24 285. (-4 2 )2 10. (3 5 )2B. 16. (2 3 - 7 )(2 3 + 7 ) 11. 2 5c . 5 5 17. ( 1 + x + 2 )2 12. 2 5 (5 3 + 3 5 ) 18. 3 ( 2 3 - 3 2 ) 13. (2 5 -4)( 2 5 + 4) 19. 3 2 ( 2 – 4)+ 2 (5 - 2 ) 14. (3 3 - 2 ) ( 2 + 3 ) 20. ( x + 3 )2 15. ( 3 + 2) ( 3 -5) C. What’s the Message? Do you feel down with people around you? Don’t feel low. Decode themessage by performing the following radical operations. Write the wordscorresponding to the obtained value in the box provided for. 9

are not 2 ·5 8and irreplaceable 37 ·47consider yourself 4 3·3 3Do notEach one 9· 4for people 3 9xy2 · 3 3 3x4 y6is unique (4 3a3 )2more or lessnor even equal 3 · 3 18of identical quality 27 · 3to others a ( a3 – 7) 5 7 ·2 7 ( 5a )( 2a (3 10a2 ) 6 36 9 a2-7 a 30a248a3 20 70 20 9xy2 3 x2 y2 6 7776 12 823543 10

Lesson 2 Division of radicals Division of radicals follows several rules, one is simplifying byrationalization of denominators. See the examples and apply them in theexercise that follows.Examples:1. Simplify: 72 6 Solution: You are given two solutions:a. Simplify 72 b. Make one radical expression72 = 36 ⋅ 2 72 = 7266 66=6 2 Rationalize = 12 6= 6 2⋅ 6 =4 3 66= 6 12 =2 3 6= 12=4 3 =2 3Note: The second method is more efficient. If you have the quotient of tworadical expressions and see that there are common factors which can bereduced, method 2 is a better strategy. 11

2. 6b7 6b7 = 6b7 Reduce 30ab 30ab 30ab Rationalize denominator Solution: = b6 5a = b2 5a = b2 ⋅ 5a 5a 5a = b2 5a 5aRationalizing Binomial DenominatorsThis would involve conjugates of denominators.3. 2 7− 5 Solution: the denominator is 7 - 5 . Multiply the numerator and thenumerator its conjugate, 7 + 5 . 2=2 ⋅ 7+ 5 7− 5 7− 5 7+ 5 = 2( 7 + 5) ( 7)2 − ( 5)2 = 2( 7 + 5) Simplify 7−5 = 2( 7 + 5) 2 = 7+ 5 12

4. 20 10 + 6Solution: 20 = 20 ⋅ 10 − 6 10 + 6 10 + 6 10 − 6 ( )= 20 10 − 6 10 − 6 ( )= 20 10 − 6 4 = 5( 10 − 6) or 5 10 − 5 65. Simplify: 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 45 and denominator which = 2(3 + 5 ) - 2 5 are not radicand. =6+2 5 -2 5 Combine similar =6 radicands. 13

6. Simplify: 12 + 18 6 Solution: Begin by simplifying the radical. 12 + 18 = 12 + 9 ⋅ 2 66 = 12 + 3 2 Factor out the common factor 6 of 3 in the numerator. = 3(4 + 2) Simplify 6 = 4+ 2 27. 2 ÷ 3 2Solution: 2 ÷ 3 2 = 2 32 1 = 22 Change the radicals to fractional exponent. Change to similar fractional exponents 1 23 3 = 26 2 26 = 6 23 Transform the expression as a single radical 22 and simplify. =62 14

8. 4xy2z2 ÷ 6 16xy2z4Solution: 4xy2z2 ÷ 6 16xy2z4 Transform to fraction= 4xy2z2 Change to fractional exponent 6 16xy2 z4 Change to similar fractional exponents 1= (4xy2z2 )2 1 (16xy2z4 )6 3= (4xy2z2 )6 1 (16xy2z4 )6= (4xy2z2 )3 Rewrite as radical expressions the radicand to powers. 6 16 xy 2 z 4 Simplify.= 64x3 y6 z6 6 16 xy 2 z 4= 6 4x2y4z29. Perform: 2 ÷ (2 + 3 ) Solution:2 ÷ (2 + 3 ) = 2 Rewrite the expression (2 + 3) = 2 ⋅ (2 − 3) Rationalize (2 + 3) (2 − 3) = 2 2− 6 Simplify 4−3 15

=2 2 - 610. Simplify: xy ÷ ( x − y )Solution: xy Rewrite the expression ( x − y) = xy ⋅ ( x + y ) Rationalize the denominator ( x − y) ( x − y) x2 y + xy2 Simplify = x− y = x y+y x x− yTry this outA. Divide and simplify 1. 6 18 ÷ 12 40 2. 8 19 ÷ 4 38 3. 20 3 ÷ 5 3 4. 42 6 ÷ 3 6 5. -4 20 ÷ 2 6. 10 18 ÷ 2 9 7. 5 96 ÷ 2 24 8. 3 30 ÷ 1 15 73 9. 20 46 ÷ 5 23 16

10. 6 3 ÷ 18 11. 12 2 ÷ 2 27 12. 12 6 ÷ 1 72 4 13. 50 ÷ 125 14. 45 ÷ 400 15. 3 a2b2 abB. 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 2 ÷ 3 2 8. 1 2+ 5 9. 1 3 − 11 10. 1 3 −1 17

D. Why is tennis a noisy game?Solve the radicals by performing the indicated operation. Find the answer belowand exchange it for each letter.E 2÷ 3 P 38 ÷36 R 4 36 ÷ 4 6 4 6 − 3 21I 34 ÷ 36 7 E 6+ 5 KL 2 ÷32 3S 2 ÷ (2+ 3 ) C 5 ÷ 15 Y 3 3x2b ÷ 3 25xy2 1A 5 63 ÷ 6 7 E 3 108 ÷ 3 2 Vx 6 28 400 1E R5 A 34 20 20 46 1 A R 80 ÷ 5R 2+ 5 5 23 3 Y 10 18 ÷ 2 9 T 5 96 ÷ 2 24A 3 −1 E 25 ÷ 625 65 S 33 ÷356x 3 36 5 3 15bxy 1 53 x 7 6−7 5 4 5 2 3 6 2 2 5y 553 3 + 3 3 18 3 75 -2 3 4 7- 2546 2 3 2 27 5 42 +5 25 3 37 33 2 6 18

Let’s summarizeMultiplication and Division of Radicals: Multiplication and division of radicals simply follows the properties inalgebraic expressions:Property 1: ab = a · bProperty 2: a= a bb If b in property is not a perfect square or a divisor of a, then rationalizationof denominators is required to simplify. 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 radical init. Hence, when we rationalize a radical expression involving two terms, multiplyby an expression that would make it equal to 1, or its conjugate. What have you learnedA. Fill in the blanks. 1. If a = b2, _______is the square root of ______. 2. When no index is indicated in a radical, it is understood that the index is_. 3 3. In radical form, 1692 is written as ____ or ____ Find the simplest form of: 4. 54 5. 3 16 6. 4 64 7. 6 64 19

8. 50x7 y119. The product of (3 2 + 4) and (3 2 − 4)10. Simplify: 26 ⋅ 4 411. The combined form of 5 7 -2 28 - 3 48 is ___________.12. In simplest form, the quotient 27 = _______ 4813. In simplest , the quotient 3 135 = _____ 3 4014. In simplest form, the quotient 7 3− 215. In simplest form, the quotient 4 162x6 y7 = ____ 4 32x8 y 20

Answer KeyHow much do you know: A. 1. 36 2. 70 3. 2 35 4. 5 10 5. 10bx B. 1. 6 3 2. 3 18 3 3. 6 2 4. 2 2 - 6 5. x y + y x x− yTry this outLesson 1A. 1. 33 2. 195 3. 12 4. 24 5. 32 6. 15 21

7. 2 30 8. 3 10 9. 2 42 10. 45B. 11. 50 c12. 10 15 + 3013. 414. 2 6 + 7 or 7 + 2 615. -3 3 – 7 or -7 - 3 316. 517. 3 + x + 2 x + 218. 6 - 3 6 19. 4 - 7 2 20. 3 + x + 2 3xC.Do not consider more or nor even to others yourself less equal Is uniquefor people are not of identical each one quality and are irreplaceable 22

Lesson 2Try this out.A. 1. 3 5 20 2. 2 19 19 3. 4 4. 14 5. - 4 10 6. 5 2 7. 5 8. 9 2 7 9. 4 210. 611. 2 6 312. 8 313. 10 514. 3 5 2015. 6 ab 23

B. 1. 6 250 2. 6 243 3 3. 312 11 3 4. 12 6 5. 12 65 or 12 7776 6. 3 7. 12 2048 2 8. -2 + 5 9. 3 + 11 −2 10. 3 3 + 3 2C. Why is tennis a noisy game?6x 3 36 5 3 15bxy 1 53 x 7 6−7 5 4 5 2 3 6 2 2 5y 55EV E RY P LA Y ERRA ISE SA R AC K ET3 3 + 3 3 18 3 75 -2 3 4 7- 2546 2 3 2 27 5 42 +5 25 3 37 33 2 6 24

What have you learnedA. 1. b, a 2. 2 3. 1693 or 4826809 4. 3 6 5. 2 3 2 6. 2 4 4 7. 2 8. 5x3y5 2xy 9. 2 10. 8 4 4 11. 7 - 12 3 12. 3 4 13. 3 2 14. 21 + 14 3y4 x2y2 15. 2x 25

Module 5 Radical ExpressionsWhat 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 basicprocedures in dealing with algebraic expressions. You will constantly be usingproperties of radicals which is in the box for easy reference. __ _ _ Property 1 √ab = √a . √b _ Property 2. a = √a b √bWhat you are expected to learn 1. Recognize basic radical notation 2. 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. 4√3 . 3√3 __ 2. 5√7 . 2√7 __ 3. 2√5. √7 __ 4. 5√2 . √5 __ _ 5. (2√x2b)(5√b ) 1

B. Divide the following expressions. __ 1. √2 ÷ √3 __ 2. 3√4 ÷ 3√6 __ 3. √2 ÷ 3√2 __ 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: n√x . n √y = n√xy 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. 2

3. rewrite the product as a single radical4. Simplify the resulting radicand if necessary.Multiplying monomial radicalsRules to follow:Rule 1. If radicals to be multiplied have the same indices, follow the stepsin the examples. ___Example 1. Multiply: √2.√3.√5Solution: 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 ___ Look for the largest perfect square = √216 factor of 216, which is 36. __ _ = √36 . √6 _ = 6√6Second approach: First put each radical into simplest form. __ __ _ _ _ _ √12 . √18 = √4. √3 . √9. √2 __ = 2√3 . 3√2 Rearrange the factors. __ = 2.3√3 √2 _ = 6 √6Note that the second approach used kept numbers much smaller. Thearithmetic was easier when the radical is simplified first. _ __Example 3. Find the product: √7 . √14Solution: _ __ ____ √7 . √14 = √ 7.14 __ 3

= √98 express the radicand as product of the largest perfect square factor. __ _ = √ 49. √2 _ = 7√2 __Example 4. Multiply: a√3 . b√6Solution: __ ___ a√3 . b√6 = ab√3.6 simply multiply the radicand having the same index. __ = ab√18 express the radicand as product of the largest square factor __ = ab√9 . √2 _ = 3ab√2 ___ ____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 √6Rule 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 . 4√5 = 51/2 . 51/4 = 5½+¼ = 53/4 ___ __ 4

= 4√53 or 4√125 ____ ____Example 7. (4√2x – 1) ( 3√2x – 1 Solution: _____ _____ (4√2x – 1 ) ( 3√2x – 1) = (2x -1 )1/4 (2x – 1)1/3 = (2x – 1) ¼ + 1/3 = (2x – 1) 7/12 _______ = 12√(2x – 1)7Rule 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 3√3 Solution: _ _ √2 3√3 = 21/2 . 31/3 = 23/6 . 32/6 __ __ = 6√23 . 6√32 ____ = 6√ 8 . 9 __ = 6√72 __Example 9. 4√2 . 3√5 Solution: _ _ 4√2 . 3√5 = 21/4 . 51/3 = 23/12 . 54/12 __ __ = 12√23 . 12√54 _ ___ = 12√8 . 12√625 _____ = 12√ 5000 Multiplying a radical by a binomial In each of the following multiplication, you are to use the distributiveproperty to expand the binomial terms. 5

_ __Example 10. Multiply: √3 ( 2√3 + √5)Solution: Using the distributive law, then __ _ _ _ __ √3 (2√3 + √5) = √3 . 2√3 + √3 . √5 _ _ ____ = 2√3.√3 + √ 3. 5 ___ = 2.3 + √15 __ = 6 + √15 __ _Example 11. Multiply and simplify: 2√x (√x - 3) – 4(3 - 5√x)Solution: Proceed as if there are no radicals- using the distributive law toremove the parentheses;__ _ __ _ _2√x (√x - 3) – 4(3 - 5√x) = 2√x √x - 6√x – 12 + 20√x __ = 2 x -6√x – 12 + 20√x __ = 2x - 6√x–12 + 20√x combine like terms _ = 2x + 14√x – 12 Binomial Multiplication. This method is very much similar to the FOIL method. The terms areexpanded by multiplying each term in the first binomial by each term in thesecond binomial. ___ _Example 12. (4√3 + √2) (√3 -5√2 ___ _ Solution: (4√3 + √2) (√3 -5√2) Use the FOIL method, that is multiplying the first terms, outer terms, inner terms and the last terms. _ _ _ _ __ _ _ = 4(√3)(√3) -4√3(5√2) + √2(√3) - √2(5√2) _ __ _ = 4(√3)2 - 20√6 + √6 -5(√2)2 __ = 4 . 3 - 20√6 + √6 – 5 . 2 __ = 12 - 20√6 + √6 – 10 6

_= 2 - 19√6 _ __ _Example 13. (√a + √3) (√b + √3) _ __ _ Solution: (√a + √3) (√b + √3) FOIL these binomial then simplify. __ _ _ _ _ _ = √a√b + √3√a + √3√b + (√3)2 __ __ __ = √ab + √3a + √3b + 3 __Example 14. Multiply and simplify: (√7 - √3 )2Solution. Watch out! Avoid the temptation to square them separately.Remember: (a+b)2 ≠a2+ b2__ _ ___(√7 - √3)2 = (√7 - √3) (√7 - √3)__ _ _ _ _ __= √7 √7 - √7 √3 - √7 √3 + √3 √3__ __= 7 - √21 - √21 + 3 Combine like terms __= 10 - 2√21_ ____Example 15. (√a – 3)2 – (√a – 3 )2Solution: 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 __ _ _ = √a√a - 3√a -3√a + 9 – (a - 3)Note that the parentheses around a – 3 is essential. _ = a – 6 √a + 9 – a + 3 _ = -6√a + 12 7

Multiplying Conjugate Binomials The product of conjugates are always rational numbers. The product of apair of conjugates is always a difference of two squares (a2 – b2), multiplication ofa radical expression by its conjugate results in an expression that is free ofradicals. __ __ Example 16. (√13 -3) (√13 + 3)Solution: Multiply out using FOIL.__ __ __ __ __ __(√13 -3) (√13 + 3) = √13 √13 + 3√13 - 3√13 – 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 __ original integer = (√5)2 – (√7)2 = -2 Simplified _ __ _Example 18. (√7 + 2√3)(√7 - 2√3) __ = (√7 )2 – (2√3 )2= 7 – 12= -5 8

Try this outPerform the indicated operations. Simplify all answers as completely as possible.A. _ __ _ __1. √3√11 6. √5 √45 _ _ __ _ _ __2. √3√5√13 7. √2√6√10 __ ___3. √6 √24 8. √3 √5 √6 __ __ __ __4. √18 √32 9. √24 √28 _ _5. (-4√2 )2 10. ( 3√5 )2B. _ _ __ __ _ 16. (2√3 - √7)(2√3 + √7)11. 2√5c . 5√5 __ _ ____12. 2√5 (5√3 + 3√5) 17. ( 1 + √x + 2 )2 __ ___13. (2√5 -4)( 2√5 + 4) ____ 18. √3 ( 2√3 - 3√2)14. (3√3 - √2) ( √2 + √3) __ __ __ 19. 3√2 (√2 – 4)+ √2 (5 - √2)15. (√3 + 2) (√3 -5) __ 20. (√x + √3 )2C. What’s Message? Do you feel down with people around you? Don’t feel low. Decode themessage by performing the following radical operations. Write the wordscorresponding to the obtained value in the box provided for. 9

are not __and irreplaceable √2 . 5√8consider yourselfDo not __Each one 3√7 . 4√7for peopleis unique __more or less 4√3 . 3√3nor even equalof identical quality __to others √9 . √4 ____ _____ 3√9xy2 . 3 3√3x4y6 (4√3a3)2 _ __ √3 . 3√18 __ _ √27 . √3 __ √a (√a3 – 7) __ 5√7 . 2√7 __ __ ___ (√5a)(√2a)(3√10a2)6 36 9 _ 30a2 a2-7√a48a3 20 70 20 ___ ___ 9xy2 3√x2y2 6√54 _ 12√7 10

Lesson 2 Division of radicalsDividing a radical by another radical, follows the rule similar tomultiplication. When a rational expression contains a radical in its denominator,you often want to find an equivalent expression that does not have a radical inthe denominator. This is rationalization. Study the following examples. __Example 1. Simplify: √72 √6Solution: 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 = 2√3 __ = 6√12 6 __ = √12 __= √4 √3 _= 2√3Note: Clearly the second method is more efficient. If you have the quotient oftwo radical expressions and see that there are common factors which can bereduced, it is usually method 2 is a better strategy, first to make a single radicaland reduce the fraction within the radical sign. then proceed to simplify theremaining expression.___Example 2. √6b7_√30ab ___Solution: √6b7__ = 6b7 Reduce √30ab 30ab 11

b6 = 5a _ = √b6 √5a = b3 √5a __ = b3 . √5a √5a √5a __ = b3√5a 5a Rationalizing binomial denominators The principle used to remove such radicals is the familiar factoringequation. If a or b is square root, and the denominator is a + b, multiply thenumerator and the denominator by a – b and if a or b is a square root and thedenominator is a + b, multiply the numerator and the denominator by a – b. (a +b) (a – b) = a2- b2Example 3. ___2___√7 - √5 __ __Solution: the denominator is √7 - √5, is the difference, so multiply thenumerator 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 + √5Example 4. ___20___ √10 + √6 __ _ Solution: ___20___ = ___20___ . √10 - √6 √10 + √6 √10 + √6 √10 - √6 12

__ _ = 20(√10 - √6) 10 – 6 __ _ = 20(√10 - √6 ) 4 __ __ __ = 5(√10 - √6) or 5√10 - 5√6Example 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 . √53 - √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 45 and denominator which __ are not radicand. = 2(3 + √5) - 2√5 Combine similar radicands. __ = 6 + 2√5 - 2√5 =6 __Example 6. Simplify: 12 + √186Solution: Begin by simplifying the radical.__ _ _12 + √18 = 12 +√9 √266 _= 12 + 3√2 Factor out the common factor 6 of 3 in the numerator. _= 3(4 + √2) simplify 6 __= 4 + √2 or 2 + √2 22 13

__Example 7. √2 ÷ 3√2__ _Solution: √2 ÷ 3√2 = __√2__ 3√2 = 21/2 Change the radicals to fractional exponent. 21/3 = 23/6 Change the fractional exponents to similar 22/6 fractions = 6 23 Transform the expression as a single radical. 22 and simplify. _ = 6√2 _____ _______Example 8. Express as a single radical: √4xy2z2 ÷ 6√16xy2z4 _____ _______ Solution: √4xy2z2 ÷ 6√16xy2z4 Transform to fraction _____ = __√4xy2z2__ 6√16xy2z4 = (4xy2z3)1/2 Change to fractional exponent (16xy2z4)1/6 = (4xy2z3)3/6 Change the fractional (16xy2z4)1/6 exponent to similar fractions. _______ Rewrite as radical expressions = 6√(4xy2z2)3 the radicand to powers. 6√16xy2z4 = 6 64x3y6z6 Simplify. 16xy2z4 = 6 4x2y4z2 __Example 9. Perform: √2 ÷ (2 + √3) __ _Solution: √2 ÷ (2 + √3) = __√2__ rewrite the expression 2 + √3 14

__ = __√2__ . 2 -√3 rationalize 2 + √3 2 - √3 __ = 2√2 - √6 simplify 4-3 __ = 2√2 - √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 ___ ___ = √x2y + √xy2 x–y __ = x √y + y√x x–yTry this outA. Divide and simplify __ __ 1. 6√18 ÷ 12√40 __ __ 2. 8√19 ÷ 4√38 __ 3. 20√3 ÷ 5√3 __ 4. 42√6 ÷ 3√6 __ _ 5. -4√20 ÷ √2 __ 6. 10√18 ÷ 2√9 __ __ 7. 5√96 ÷ 2√24 __ __ 8. 3/7 √30 ÷ 1/3 √15 __ __ 9. 20√46 ÷ 5√23 15

_ __10. 6√3 ÷ √18 _ __11. 12√2 ÷ 2√27 _ __12. 12√6 ÷ ¼ √72 __ ___13. √50 ÷ √125 __ ___14. √45 ÷ √40015. 3 3x2b ÷ 4 25xy2B. 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 + √5 9. __1__ 3 - √11 10. __3__ √3 – 1D. Why is tennis a noisy game?Solve the radicals by performing the indicated operation. Find the answer belowand exchange it for each radical letter. 16

__ __ __ _E √2 ÷ √3 P 3√8 ÷ 3√6 R 4√36 ÷ 4√6 __ ___7___ _ __I 3√4 ÷ 3√6 E √6 + √5 K 4√6 - 3√21 __ _ __ √3 ____ ____L √2 ÷ 3√2 C √5 ÷ √15 Y 3√ 3x2b ÷ 3√25xy2 __ V _1_ ___ _S 2√2 ÷(2+√3) √x E 3 √108 ÷ 3√2 __ __ A 400 R _1_A 5√63 ÷ 6√7 20 __ _ √5 __ __ R √80 ÷ √5E 6√28 A 20√46 __ 3√4 5√23 Y 10√18 ÷ 2√9 __ __R __1__ __ ___ T 5√96 ÷ 2√24 2+√5 E √25 ÷ √625 65A __3__ __ √3 - 1 S 3√3 ÷ 3√5√6 √x _ 3√36 _______ __32 7 4 5√2 3 6√2 5 3√15bxy √5 √5 2 5y 5 5_ 3√3+3 3√18 _ 3√75 _ _ _ √3 _ _ _ 254√ 6 2 3 2 2√7 5 4√2 -2+√5 2√5 3 4√7-3√7 3 3√2 6Let us summarize 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. 17

What have you learnedA. 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 3√16 is ____ __ 6. In simplest form 4√64 is ____ __ 7. in simplest form, 6√16 is ___ _____ 8. In simplest form √50x7y11 ____ ____ 9. The product of (3√ 2 + 4)(3√2 – 4) __ __ 10. The product √26 . 4√4 _ __ __ 11. The combined form 5√7 -2√28 - 3√48 is ___________. __ 12. In simplest form, the quotient √27 = _______ √48 ___ 13. In simplest , the quotient 3√135 = _____ 3√40 _ 14. In simplest form, the quotient __√7__ √3 - √2 ______ 15. In simplest form, the quotient 4√162x6y7 = ____ 432x8y 18

Answer KeyHow much do you know: A. 1. 3.4.3 = 36 2. 5.2.7 = 70 __ 3. 2√35 4.5√10 5. 10bx _ B. 1. √6/3 __ 2. 3√18/3 _ 3. 6√2 __ 4. 2√2 - √6 __ 5. x√y + y√x x-yTry this outLesson 1 __A. 1. √33 ___ 2. √199 3. 12 __ 4. 12√12 5. 32 6. 15 _ 7. 2√3 __ 8. 3√10 __ 9. 2√42 19

10. 45 _B. 11. 50√c __ 12. 10√15 + 30 13. 4 __ 14. 2√6 + 7 or 7 + 2√6 __ 15. -3√3 – 7 or -7 - 3√3 16. 5 ____ 17. 3 + x + 2 √x + 2 _ 18. 6 - 3√6 _ 19. 4 - 7√2 __ 20. 3 + x + 2√3xC.Do not consider more or less nor even to others yourself equalfor people are not of identical each one Is unique quality and are irreplaceable 20

Lesson 2 _ 11. 2√6Try this out. _ 3 __A. 1. 3√5 12. √12 20 4 __ __ 13. √10 2. 2√19 5 19 _ 3. 4 14. 3√5 4. 14 _____ 15. 4√75bxy2 __ 5. - 4√10 5y _ _ 6. 5 √2 8. -2 +√5 7. 10 __ _ 9. 3 + √11 8. 9√2 -2 2 _ _ 10. 1 + 3√3 9. 4√2 2 _ 2110. √6 ___B. 1. 6√250 ___ 2. 6√243 3 __ 3. 12√311 3 _ 4. 12√6 __ ____ 5. 12√65 or 12√7776 _ 6. √3 ___

7. 12√243C. Why is tennis a noisy game?√6 √x _ 3√36 _ 5 3√15xby √5 √5 5√2 3 6√2 6 5y 5 532 74 A YPL YE REVERRA I SE S A R AC K ET _ _ _ 25_ 3√3+3 3√18 3√75 _ _ √3 4√2-3√7 3 3√2 64√6 2 3 2 2√7 5 4√2 -2+√5 2√5 3What have you learnedA. 1. a,b 2. 2 ____ ________ 3. √169 3 or √ 4826809 _ 4. 3√6 _ 5. 2 3√ 2 _ 6. 2 4 √4 __ 7. 6√64 ___ 8. 5x3y5 √2xy 9. 18 + 9x 10. 6 √16 __ 11. √7 - 12√3 12. 3/4 13. 3/2 __ __ 14. √21 + √14 ______ 15. 34√72x2y6 4x 22

Module 5 Searching for Patterns in Sequences, Arithmetic , Geometric and others What this module is about This module is about how geometric sequences are formed. Here you willdetermine the common number that is multiplied to a number to get the next term in thesequence. As with arithmetic sequences, which you have just finished, there is aformula that will aid you in answering the problems given in this module. What you are expected to learn It is expected that you will be able to demonstrate knowledge and skill related togeometric sequences and apply these in solving problems. Specifically, you should be able to: a. given the first few terms of a geometric progression, find the: · common ratio · nth term b. given 2 specified terms of a geometric progression, find the: · first term · common ratio c. solve problems involving geometric means.

How much do you know A. Which of the following is a geometric sequence? Write GP, if it is or not, if not. 1. 1, 1 , 1 , 1 ,… 3 9 27 2. 6, -2, 2 , - 2 ,… 39 3. 100, 10, 1, 0.01,0.0001,… 4. 1, 1.1, 1.21, 1.331,… 5. 2, 4, 6, 8,… B. Do as directed. 6. Find the common ratio of the geometric sequence 5, 15, 45, …. 7. What is the general term for the geometric sequence in no. 1? 8. Find the 8th term of the sequence 5, 10, 20, 40,…. 9. Find the first 6 terms of a geometric sequence with a1= 81, r = 1 . 3 10. Insert 3 geometric means between 7 and 112. What you will do Lesson 1 Geometric Sequences By this time, you are already familiar with arithmetic progressions or sequences.In an arithmetic progression, each term after the first could be found by adding aconstant term to the previous term. But in this lesson, we are concerned with sequencesthat can be obtained by multiplying each term by a constant number. Such sequencesare called geometric progressions or sequences. The constant number is called thecommon ratio. 2