Math 10 Module 1 PDF

Poveda Remote, Independent, & Self-Paced Education (RISE) Home School Program Math 10 Module 1. Rational Expressions, and Laws of Radicals, and Applications of Rational Expressions Duration: 4 Weeks...

Poveda Remote, Independent, & Self-Paced Education (RISE) Home School Program Math 10 Module 1. Rational Expressions, and Laws of Radicals, and Applications of Rational Expressions Duration: 4 Weeks This module covers a recall of the Laws of Exponents, Applications of Laws of Exponents to expressions with fractional exponents, transforming expressions with fractional exponents to radical forms, simplifying radicals and performing operations involving radicals; and Solving equations with radical expressions. Individual Work for Module 1: To answer Part I of this Activity Card, you have to truly understand the true meaning of a person considered to be an “exponent” of something and be aware of the person’s acts that manifest this characteristic. Before doing Part II, it is strongly advised to first go through the main discussions and study all rules in working with expressions with non-integral rational exponents as well as those in radical forms Click here to start working on your Individual Work Output 1 TOPIC 1. LAWS OF EXPONENTS AND THEIR APPLICATIONS IN EXPRESSIONS WITH FRACTIONAL EXPONENTS By the end of this topic, you will: a) know: 1. the difference between expressions with integral exponents and those with non-integral rational exponents; 2. how the laws/rules of exponents are applied in expressions with non integral rational exponents; 3. the definition of an expression with zero exponent; and 4. the form of an expression with negative exponent when transformed to one with a positive exponent. b) be able to: 1. define an expression with zero exponent, with negative exponent; 2. apply the laws of integral exponents in expressions with non-integral rational exponents; and 3. transform expressions with negative exponents to a form with positive exponents. 2 PART I: REVIEW OF EXPONENTS Can a person like any of us be an exponent? The word exponent is used not only in mathematics but in non-mathematical settings as well, where it is used to refer to a person who is a supporter of, a strong believer of, or an advocate of something. A person for example who strongly believes in the principle of individual differences can be referred to as an exponent of that principle. St. Pedro Poveda is an exponent of this principle. Other terms synonym to the term exponent are: promoter, champion, defender, backer, upholder. Think of what you are an exponent of or what you want to become an exponent of. In mathematics however, an exponent refers to the number of times a number or expression should be taken as a factor.  You met exponents before in your previous Math subjects and hence you know that an exponent is placed at the upper right-hand side of a number or expression, written a bit smaller in size than the number/expression. 53is read as five cube or five to the 3rd power, which means that 5 is taken 3 times as a factor, as (5)(5)(5). Another way of describing an exponent is, it is a number to which a number or expression is raised. So, 53 may also be read as, 5 is raised to 3 or to the third power or to the 3rd degree, and is also called a power of 3. In fact, any number or expression raised to an exponent is called a power of that particular exponent. If the exponent is 2, like 2, 72, ( + 2)2, etc., it is read as “b squared, 7 squared, x+3 squared; or b to the 2nd power, 7 to the 2nd power, etc.; or b raised to 2, 7 raised to 2, etc. If the exponent of these bases is 3, it is read as b cube, 7 cube, x+3 cube, etc.; or b to the 3rd power, 7 to the 3rd power, etc.; or b raised to 3, 7 raised to 3, etc. Any exponent higher than 3 is read as to the 4th power, to the 5th power, etc.; or raised to 4, raised to 5, etc. If the exponent is a variable or an expression, it is read as 2 to the x, or x to the n, or b to the xz, etc.; or raised to x, raised to xz, etc. 3 PART II: REVIEWING THE LAWS OF EXPONENTS Let us recall laws or rules that need to be observed or followed in working with numbers or expressions with exponents.  1) Multiplication Rule/ Product Rule: Product Law ( )( ) = + Examples: ( 5)( 4) = 5+4 = 9 (74)(7) = 74+1 = 75 Take note that the rule requires that the bases of the expressions to be multiplied (or the factors) must be the same for the rule to be applied. In the case of (43)(72) or 4 6, or in any expression where the bases of the factors are not the same, this rule cannot be applied. 2) Division Rule/ Quotient Rule: Quotient Law ( ) ÷ ( ) = = + Examples: 12 ÷ 7 = 12 7 12−7 = = 5 5 78 ÷ 75 =787 = 78−5 = 73 Like the Multiplication Rule, this rule also indicates that it is only applicable if the dividend and the divisor have the same base. So, in 153 ÷ 52, or m8 ÷ n4, or in any instant when the bases of the dividend and the divisor have different bases, it is not right to apply this rule. 3) Power Rule; Power Law ∗ ( ) = Examples: ( 4)3 = 4∗3 = 12 (62)5 = 62∗5 = 610 4 4) Power of product Power of Product Law ( ) = Examples: ( )3 = 1∗3 1∗3 = 3 3 (7 )2 = 72 2 = 49 2 When raising a product of powers, to an exponent, use the power rule and power of product rule together. ( ) = ( ) ( ) = ∗ ∗ Examples: ( 5 3)2 = ( 5)2( 3)2 = 5∗2 3∗2 = 10 6 [23 2]4 = (23)4( 2)4 = 23∗4 2∗4 = 212 8 5) Power of Quotient Power of Quotient Law ( ) = Examples: (23)4=24 4 3 =16 81 (5 )2=52 2 =25 2 When raising a quotient of powers, to an exponent, use the power rule and power of product rule together. ( =( )  ( ) = ∗ ) ∗ Examples: 2 5 (3)(5) )= 3 ( ) (2)(5 = 15 10 7 3 4(2)(3) )= 42 ( (7)(3) =46 21 =4 096 21 5 Two other things that we have to recall are the: The definition of an expression with 0 Zero Exponent Law 0 exponent = 1 Any expression raised to 0 is equal to 1. “x to the 0 is 1”. Examples: 0 = (24)0 = 1 (−12 )0 = 1 (613)0= 1  The definition of an expression with negative Negative Exponent Law − exponent 1 1 − = = any expression with a negative exponent is equal to its reciprocal with a positive exponent any x-n = 1 , ≠ 0. This is read as, x to the negative n, is equal to (or the same as), 1 over x, raised to the n. Another way to say this is: Any number or expression raised to a negative exponent, is the same (or is equal to), and can be written as, its reciprocal but with a positive exponent. Examples: −4 =1 4 −2 1 17 = 2 17 8 −3 ( 9) =1 3 8 ( 9) = (98)3 6 The following are illustrations of the two definitions. Expression 1: =? By applying the division rule above, we have: 5 5 = 5−5 = 0 Another way of showing this is:  5 5 =( )( )( )( )( ) ( )( )( )( )( )= 1 This shows that y0is indeed = 1. Expression 2: =? By applying the rule in division, we have: 5 7 = 5−7 = −2 by definition of an expression with negative exponent 1 −2 = 2 Also 5 7 =( )( )( )( )( ) ( )( )( )( )( )( )( )=1 This shows that 5 ( )( 1 2 7 2 )= -2 1 = y =. For practice in applying the rules for exponents above, accomplish the following worksheet. For maximum learning of the processes, please click on the answers only after you have worked on the items. Prodigy (35 items) 7 PART III: EXPRESSIONS WITH NON-INTEGRAL RATIONAL EXPONENTS AND RULES Video on converting radicals to fractional exponents and vice versa Recall that Radicals can be rewritten as expressions with fractional exponents. We simply write the index of the radical as the denominator of the exponent = √ It is interesting and good that all the above rules and two definitions are applicable not only with expressions whose exponents are integers but even with expressions whose exponents are fractions (also referred to as non-integral rational exponents). Below are only few illustrations:  a) Multiplication Rule: b) Division Rule: 3 3 ∗ √ 2 4 ÷ √ 2 4 √ 3 √ 3 = ( 34) ( 23) = ( 34) ÷ ( 23) = 34+23 = 4 3 [ 34 and 23 are read as, x to the 23 three fourth and x to the two-third.] = 34−23 = 912+812 = 912 −812 = 17 = 112 12 8 c) Raising a product of powers to d) Definition of an expression an exponent: with negative exponent: 1 4 4 3 )] √ −3 ) (√ 2 4 −3 [(√ 3 = 4 = [( 34) ( 23)]14 =1 (3 )(1 (2 )(1 = ( 4 4)) ( 3 4)) 34 = ( 316) ( 212) = ( 316) ( 16) This link from varsity tutors provides items for practice in applying the Laws of Exponents as applied in expressions with fractional exponents. Work on the items first before you click on the answers. 9 SELF -EVALUATION Before moving to the next topic, check how much you learned by answering the following questions: 1. How do you explain in words, the five Laws of Exponents? 2. With what forms of expressions are the rules of exponents for multiplication and division only applicable? 3. How can an expression with a negative exponent be written with a positive exponent? 4. Is it possible for an expression with an exponent of 0 to be equal to zero? Briefly explain your answer and give illustration or an examples. 5. Which of the five Laws do you most appreciate and why? 6. Which group of expressions with exponents do you manipulate more efficiently, those with integral exponents or those with non-integral rational exponents? Briefly explain your choice. 7. In the range of 1 to 5, how much have you learned this topic? 8. What do you plan to do to be more proficient in the topic?  REFERENCES/SOURCES: Text Orines, F., Mercado, J., Suzara, J., & Manalo, C. (2012). Next Century Mathematics. Phoenix Publishing House. Huettenmueller. R. (2021). Algebra DeMYSTiFieD 2nd Second edition byHuettenmueller (2nd Second edition). McGraw-Hill Professional. Websites Pierce, Rod. (14 Aug 2021). "Fractional Exponents". Math Is Fun. Retrieved 8 Nov 2022 from http://www.mathsisfun.com/algebra/exponent-fractional.html Pierce, Rod. (14 Sep 2021). "Laws of Exponents". Math Is Fun. Retrieved 8 Nov 2022 from http://www.mathsisfun.com/algebra/exponent-laws.ht Images 387098. (n.d.). Clipart Library. http://clipart-library.com/clipart/387098.htm Exponent Cliparts #100768. (n.d.). Clipart Library. http://clipart library.com/clipart/387098.htm 10 Videos AccokeekProgramming. (2015, February 15). Power of a Quotient Property [Video]. YouTube. https://www.youtube.com/watch?v=6KWwe2vdQ04 Khan Academy. (2007, January 27). Exponent rules part 1 | Exponents, radicals, and scientific notation | Pre-Algebra | Khan Academy [Video]. YouTube. https://www.youtube.com/watch?v=kITJ6qH7jS0&t=496s Mashup Math. (2015a, September 24). Exponent Rules: Multiplying Exponents with the Same Base! [Video]. YouTube. https://www.youtube.com/watch?v=7gZBCTw2EmI Mashup Math. (2015b, September 28). Exponent Rules: The Power to Power Rule! [Video]. YouTube. https://www.youtube.com/watch?v=39l-MZFUEzY Mashup Math. (2015c, October 5). Exponent Rules: Dividing Exponents with the Same  Base! [Video]. YouTube. https://www.youtube.com/watch?v=khLTbG0VB3Q Mashup Math. (2015d, October 5). Exponent Rules: Product to a Power Explained! [Video]. YouTube. https://www.youtube.com/watch?v=kbOqDoWVJmE Mashup Math. (2017, January 4). Multiplying Negative Exponents Using the Negative Exponent Rule! [Video]. YouTube. https://www.youtube.com/watch?v=wQmtsgRMGmU Simple Math. (2017, March 15). Why does “x to the zero power” equal 1? [Video]. YouTube. https://www.youtube.com/watch?v=yiwAS3R-mG0 The Organic Chemistry Tutor. (2017, May 6). Simplifying Exponents With Fractions, Variables, Negative Exponents, Multiplication & Division, Math [Video]. YouTube. https://www.youtube.com/watch?v=Zt2fdy3zrZU&t 11 TOPIC 2. EXPRESSIONS WITH NON-INTEGRAL RATIONAL EXPONENTS AND THEIR RADICAL FORMS, LAWS OF RADICALS AND OPERATIONS ON RADICAL EXPRESSIONS, SOLVING EQUATIONS WITH RADICALS By the end of this topic, you will: a) know: 1. the radical forms of expressions with non-integral rational exponents; 2. the laws/rules of radicals which are followed in working with radical expressions. b) be able to:  1. perform algebraic operations involving expressions with non-integral rational exponents; 2. transform expressions with non-integral rational exponents to radical forms and vice-versa; 3. simplify radical expressions; 4. perform algebraic operations involving expressions with radicals; and 5. solve equations with radical expressions. 12 PART I: REVIEWING RADICALS Can expressions with fractional exponents be written in another form? Yes, they can. By definition, expressions with fractional exponents may be written in radical forms. A dictionary defines the word radical as a growth from the root/s or something that is extreme or out of the ordinary. In mathematics I think these two meanings are adapted because in mathematics, is called the nth root of x (or of whatever it is that is raised to 1n), and is written 1 in , where √⬚ is called the radical sign, is called the an extra ordinary way as, √ radicand, and is called the index or order of the radicand. This index indicates the number/s or root/s of the radicand, which when taken n number of times as a factor, is equal to the radicand. (The images below, illustrate the radical expression in detail.) As the above illustrations show, given the radical expression √1 024 5 It is read as, 5th root of 1024 Its index is 5 Its radicand is 1024 is read as, the nth root of x. √ The root/s of this is a number or are numbers or expression/s such that when taken n-times as a factor, is equal to x. For instance, the cube root of 27 (which is written as √27 3) is 3. It is because (3)(3)(3) = 27. The cube root of -27 or √−27 3is -3 because (−3)(−3)(−3) = − 27. 13 The fourth root of 16, written as √16 4, has a principal (positive) root 2 and has a secondary (negative) root -2. This because (2)(2)(2)(2) = 16 and (−2)(−2)(−2)(−2) = 16. Normally, only the principal root is given, unless the secondary root is required, like if what is asked is −√16 4. The root of this is -2. If what is asked is ±√16 4, then both roots are given, i.e., ±2. Normally, if the index is even and the radicand is positive, only the principal or positive root is considered. That is why in some instances, the root of a positive = │c│, where a radicand whose index is even is written as an absolute value, like, √ > 0, b is  even, and c is the number or root, which, when raised to b, is a. In symbols, = │c│, because c b = a. this is √ As mentioned above, the negative root is only considered when it is asked or specified, like when there is a negative or ± sign before the radical expression. This is illustrated in Example 1 below. If the index is odd and the radicand is negative, like √−125 3, (read as, the cube root of negative 125), the only answer is -5 since (−5)(−5)(−5) = −125. The number 5 does not satisfy this. If the index is even and the radicand is negative, there is no real root or real number that can satisfy it. For example, the fourth root of -16, written as, √−16 4, is neither 2 nor -2, since neither of them will be equal to -16 if they are taken 4 times as a factor, Examples: 1) 8114 = √81 This is read as: o 81 to the 14 o the fourth root of 81 It has 4 as its index and 81 as its radicand. Its root is 3 because (3)(3)(3)(3) = 81. o Although -3 is also a number such that (-3)(-3)(-3)(-3) is also 81, it is not considered, since it is not specified. o The root 3 is its principal root and -3 is its secondary root. o If the expression were ±√81 4, the answer would be ±3, and if it were, - √81 4, the answer would only be -3. 14 2) 3215 = √32 5 It is read as: o 32 to the 15 o the fifth root of 32 It has 5 as its index and 32 as its radicand. Its root is 2, because (2)(2)(2)(2)(2) = 32. 3) m13 = √m 3 It is read as  o m to the 13 o the cube root of m It has 3 as its index and m as its radicand. Whatever is the value of m, we need to determine a root or number which when taken 3 times as a factor, will be equal to the value of m. )(√ )(√ o (√ o 15 to the 14 ) = 4) (15) = √15 14 4 It is read as: o the fourth root of 15 It has 4 as its index and 15 as its radicand. We need to look for a root or number which when taken 4 times as a factor, will be equal to 15. o (√15 4)(√15 4)(√15 4)(√15 4) = 15 o The answer to this is a non-integer just a bit less than 2. ▪ A scientific calculator can give you the approximate value of 1.968. 5) (64 12 18)16 = √64 12 18 6 This is read as o the 6th root of the quantity 64, x to the 12, y to the 18. o Its radical form has 6 as its index and 64 12 18 as its radicand. o Its root is 2 2 3 because (2 2 3)(2 2 3)(2 2 3)(2 2 3)(2 2 3)(2 2 3) = 64 12 18. 6) 12 = √ 2 , Can be written as √ This is read as o p to the ½ o the square root of p o It has 2 as its index and p as its radicand. 15 To practice, do all the items provided in this link. It provides you with each solution once you click on your answer. PART II: REWRITING RADICALS AS FRACTIONAL EXPONENTS. Video on simplifying fractional Video on converting radicals to exponents fractional exponents and vice versa Recall that Radicals can be rewritten as expressions with fractional exponents. We simply write the index of the radical as the denominator of the exponent = √ We can also use the power rule to help us rewrite the radical, whose exponent is greater than 1, into different forms. For Example 35 can be rewritten as √ 3 5 By reversing the power law, we can also rewrite 35 as 15∗3or ( 15) o ( 15) can be rewritten as (√ 3 ) In general, using the exponent rule on raising a power to an exponent, we can write any ( )1 as and vice versa. By definition, any 1 , is the nth root of whatever that is raised to 1 , that is 1 = √. = √ = 1 ∗ = ( )1 = (√ ) = √ = (√ ) 16 Example 1. 823 can be written and simplified as: 3 )2 √82 (√ 3 = √8 ∗ 8 )2 = √64 3 = (√2 ∗ 2 ∗ 2 = 22 3 = √4 ∗ 4 ∗ 4 =4 =4 Example 2. ( )32 can be written and simplified as: 2 3 (√ 2)3 √( ) 2 √ 3y3 3 (√ 2)(√ 2 2 )(√ 2) √ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ √ 2 2 ∗ ∗ √ ∗ ∗ [In case you are wondering why √ (√ )(√ = , note that 2 (√ )(√ ) = ( )12( )12 = ( )12+12 = ( )1 = As shown by the two solutions above, either way of writing the radical form of the expression will give the same answer. This link provides another way of presenting how to work with expressions with varied exponents and also more items to work on. 17 Let’s practice by answering the exercise below. Exercise 1: Simplifying expressions with Fractional Exponents Instructions: Simplify the following expressions. ( 8)32 (9 4)0.5 (81 12)1.25 (216 9)13  − (225 6 8)−2.5 ( ) Click here to check your answer 18 PART III: LAWS OF EXPONENTS ON FRACTIONAL EXPONENTS. It is interesting and good that the rules and two definitions on exponents are applicable not only with expressions whose exponents are integers but even with expressions whose exponents are fractions (also referred to as non-integral rational exponents). Below are only few illustrations: a) Multiplication Rule: b) Division Rule: 3 3 ∗ √ 4 2 ÷ √ 2 4 √ 3 √ 3 = ( 34) ( 23) = ( 34) ÷ ( 23) = 34+23 = 4 3 [ 34 and 23 are read as, x to the 23 three fourth and x to the two-third.] = 34−23 = 912+812 = 912 −812 = 17 = 112 12  c) Raising a product of powers to d) Definition of an expression an exponent: with negative exponent: 1 4 4 3 )] √ −3 4 ) (√ 2 −3 [(√ 3 = 4 = [( 34) ( 23)]14 =1 (3 )(1 (2 )(1 = ( 4 4)) ( 3 4)) 34 = ( 316) ( 212) = ( 316) ( 16) This link provides items for practice in applying the Laws of Exponents as applied in expressions with fractional exponents. Work on the items first before you click on the answers. 19 PART IV: LAWS OF RADICALS We can derive more rules/laws that can be used on radicals. A. Multiplying Radicals with the Same Index Video on Multiplying radicals with the same index If two radical expressions with the same index, are being multiplied together, multiply the radicands. = √  √ √ This derived from the Power of Product Law of Exponents. Instead of distributing the same exponent to all terms, we factor out the similar exponent. = 1 1 = ( )1 = √ √ √ Examples. (√12)(√3) 4 1 (√27 5)(√−9 (√ 75 ) (√ 3) 5 = √12 ∗ 13 ) 5 = √36 4 1 = √ 75 ∗ 3 = √27 ∗ −9 =6 = √−243 5 = √4 =−3 225 2 = 15 20 B. Dividing Radicals with the same index. If two radical expressions with the same index, are being divided by each other, divide the radicands. =√ √ √ This derived from the Power of Quotient Law of Exponents. = 12 Examples. 12= ( 12 ) = √ √ √  4 √500 3 √637 3 3 √13 √ 4 √−4 4 4 √ 27 = √500 = √637 4 3 13 3 4 = √ 4÷ 27 −4 = √49 4 = √−125 3 =7 3 27 = √( 4) ( 4) =−5 4 81 = √ 16 3 = 2 C. Simplifying a radical whose index and exponent are equal. ) = (√ Once we rewrite the expression into a fractional exponent, we can see that it is simplified to an exponent of 1. ) = = 1 = (√ Examples. (√27 5)5 (√43 8)8 )4 4 = 27 = 43 (√98 3 4 = √98 3 21 PART V: SIMPLIFYING RADICALS Video on simplifying radicals Skills in factoring and number divisibility testing can facilitate in working with radicals.  For instance, to get √729 3, it helps to know that 729 is divisible by 3, since the sum of its digits, which is 18, is divisible by 3. 3 √729 3 3 = √36 = √3(243) = 362 3 = √3(3)(81) = 32 3 =9 = √3(3)(3)(27) 3 = √3(3)(3)(3)(9) 3 = √3(3)(3)(3)(3)(3) 3 = √36 Another example is √4096 6. At first glance it may seem hard to get the 6th root of this radicand. But looking closely at it, its last two digits is divisible by 4. 6 6 = √4(1024) = √46 6 = √4(4)(256) = 466 6 =4 = √4(4)(4)(64) 6 = √4(4)(4)(4)(16) 6 = √4(4)(4)(4)(4)(4) 6 = √46 There are instances when the radicand is not a perfect power of the index, like √18. = √2 ∗ 9 = √2 ∗ √32 = √2 ∗ 3 ∗ 3 = 212 ∗ 322 = √2 ∗ 32 = √2 ∗ 3 = √2 ∗ √32 = 3√2 This kind is sometimes called a mixed radical (a product of a radical and another number). 22 Like what is done with fractions, radical expressions need to be simplified when they are not in their simplest forms. The image above clearly indicates the three things to be remembered to ensure that a radical is in its simplest form. is considered in simplest form when: A radical expression, √ a) its radicand is not a perfect (where a is any non-zero real number) nor does it have a factor which is a perfect. This means that if the index of the radical expression is 2, the radicand must not be a perfect square (a2) nor must have a factor that is a perfect square; if the index is 3, then the radicand must not be a perfect cube (a3) nor must have a factor that is a perfect cube; if the index is 4, then the radicand must not be a perfect a4 nor must have a factor which is a perfect a4 and so on. To simplify the radical expression of this kind, is to determine the root of every perfect anin the radicand and that root must be written outside the radical sign, making every an disappear inside the radical sign or in the radicand. Examples. 1) √32 3. This is not in simplest form because although its radicand, 32, is not a perfect cube, it has a factor which is a perfect cube. √32 3 3 = √8 ∗ 4 3 = √23 ∗ 22 = 233223 = 2√4 2) √200 3 2 5. This is not in simplest form for several reasons: the numerical coefficient, 200, can be factored as 100(2) and 100 is a perfect square; 2 y is a perfect square; x3 can be factored as x2(x) and x2is a perfect square z5 can be factored as z4(z) and z4is a perfect square 23 √200 3 2 5 √2(102) 2 2 4 = 212(102)12( 2)12( )12( 2)12( 4)12( )12 = 212 (1022) ( 22) 12 ( 22) ( 42) 12 = √2 ∗ 10 ∗ ∗ √ ∗ ∗ 2∗ √ = 10 2√2 4 3) √625 5 3 7 This is not in its simplest form. 625 = 54 5 = ( 4)( ) 7= 4 3 4 √625 5 3 7 4 = √54 4 3 4 3 4 = 5 √ 3 3 3. 4) √12 2 This is in simplest form, since not one of the expressions in the radicand is a perfect cube nor has a factor which is a perfect cube. b) it has no fraction in the radicand We can multiply the numerator and denominator of the radicand with the denominator to get a perfect square denominator. If the numerator and denominator of the new factor are equal, it will not change the value since it is equal to 1.  4 3 2 √ 3 2 2 5 √ √ 4 2 3 2 3 =√ 3( 3) 2 = √ ( 2 5 2 4 = √( 4) ( 2) 6 2 2 =√ 3 ) 3 10 =√ 8 =√6 2 √32 = √2 3 4 10 = √ 23 4 =√6 4 3 3 3 2 =√10 √2 =√2 3 4 4 √ 3 4 =√10 2 2 =√2 24 c) it has no radical expression in its denominator. We multiply the numerator and denominator with a root that will turn our denominator perfect. Example 1. 7√ 7 = ( √ ) (√ √ ) =7√ √ 2 =7√ 2. √5 3 √2 3 2 2 3 ) (√2 3) = (√5 √2 √22 2 3  2 2 = (√5 ∗ √2 3 ) 2 2 ∗ √2 3 √2 3 2 2 = (√5 ∗ √2 3) 3 3 √2 3 2 2 = (√5 ∗ √2 3) 3 3 ∗ √ √23 3 2 2 =√5 ∗ √2 2 We can place the radicands in the numerator by converting them into fractional exponent form, then converting the exponents to equivalent fractions with the same denominator, then using the laws of exponents and radicals to simplify it. 12 2 2 1 =5 ∗ (2 ) 6 2 12 2 2 1 =5 ∗ (2 ) 3 2 36 2 2 2 =5 ∗ (2 ) 6 2 25 6 ∗ √(22 2)2 6 3 =√5 6 ) 2 ( 6 ∗ √24 4 3 =√5 ) 6 2 ( ℎ 3 4 =√5 2 4 2 (  6 =√125 ∗ 16 ∗ 4 ) 6 2 √7 3. 5 2 3 √7 4 3 =√2000 2 5 3 3 2 5 (√7 5) =√7 √73 √72 2 3 5 ∗ √73 2 3 =√7 5 √72 373 2 5 ∗ √73 2 3 =√7 5 √75 5 5 ∗ √73 2 3 7 =√7 13 =7 ∗ (73 2)15 7 515 =7 ∗ (73 2)315  7 15 15 ∗ √(73 2)3 5 =√7 7 15 15 ∗ √79 6 15 5 =√7 7 14 =√7 6 7 26 3 4. + √7 To remove the radical expression in the denominator in expressions of this form, it requires that the entire radical expression be multiplied by 1, in the form: conjugate of the denominator over itself. [The conjugate of a binomial with a radical expression is also a binomial, but its terms are either the sum or difference of the terms of the binomial with radical expression, that is, if the binomial is + √7, its conjugate is − √7 and the other way around. If the binomial is (√3 − 8), its conjugate is (√3 + 8) and vice versa. Multiplying binomials with radical terms is like multiplying any two binomials where distributivity in multiplication over addition or subtraction is applied. That is, (4 + √5)(3 - √7 ) = 12 – 4√7 + 3√5 - √35.] In Example 4 its denominator is ( + √7). The conjugate of this is ( − √7). (3 + √7) ( − √7 − √7)  =3 ( − √7) ( + √7)( − √7) 2 =3 − 3 √7 2 + √7 − √7 − 7 2 =3 − 3 √7 2 − 7 The radical sign in the denominator then, is eliminated. This process of removing the radical sign in the denominator of a rational expression is called rationalizing the denominator, which simply means, making the denominator a rational number. 27 Now let’s it’s time to check how much you did learn. Before moving to the next Topic, make sure you have submitted the following in your online classroom! You may click the links bellow to download the activities. Module 1 Worksheet 1: Laws or Exponents, Radicals and Simplifying Radicals 28 PART VI: ADDITION AND SUBTRACTION OF RADICAL EXPRESSIONS Video on Adding and Simplifying Radical Expressions  Radical expressions that can be added are only those with the same: a) indices (or orders) and b) radicands. These kinds are called similar radicals or like radicals, just like the similar or like terms among polynomials. Given the 10 radical expressions below, group the similar ones then add/subtract them. 5 3 3 + 5√6 5 − 10√6 4 − 3√6 + √−7 3 √6 4 + 5√6 + 8√−7 4 3 − √12 + 2√6 5 4 + √−7 5 − 5√12 3− 7√6 4 − √12 3+ 2√6 4 − 5√12 3− 7√6 3 − 10√6 4 = √6 − 3√6 5 4 − 10√6 5 ) + (√−7 + 8√−7 = (√6 + (−√12 3− 5√12 3) 3 + 8√−7 5 ) + (9√−7 3 ) + (5√6 3 ) + (− 3√6 4 + 2√6 = (−9√6 4 − 3√6 5 + 9√−7 4 − 7√6 4 ) + (−6√12 3) 4− 6√12 3 4 ) 3 = −9√6 − 3√6 There are instances when radical expressions do not look similar. But in some cases, when these radical expressions are simplified (transformed to their simplest form), their being similar becomes evident. Examples. 1) √72 + 2√18 − √162 = √3 ∗ 3 ∗ 2 ∗ 2 ∗ 2 + 2√3 ∗ 3 ∗ 2 − √3 ∗ 3 ∗ 3 ∗ 3 ∗ 2 = 3 ∗ 2√2 + 2 ∗ 3√2 − 3 ∗ 3√2 = 6√2 + 6√2 − 9√2 = 3√2 2) 4√3 - √27 - 4√13 29 1 3 = 4√3 − √9(3) − 4√( 3) ( 3) 3 2 = 4√3 − 3√3 − 4√( 3 ) = 4√3 − 3√3 −4√3 √32 = 4√3 − 3√3 −4√3 3 4 = 4√3 − 3√3 − 3√3 4 = 1√3 − 3√3 1 = − 3√3 3 – 10√40 3+ 20√625 3 3 + 20√5(125) 3) 24√5 3 3 − 10√8(5) = 24√5 3 + 20√5(53) 3 3 3 − 10√2 (5) = 24√5 3 3 − 10 ∗ 2√5 = 24√5 = 24√5 3 3 + 20 ∗ 5√5 + 100√5 3 3 − 20√5 3 + 100√5 3 = 4√5 3 = 104√5 30 It is advised that the links below be visited and explored since they provide varied discussions and examples on simplifying radicals and performing algebraic operations involving radicals are provided in these links. Paul’s Online notes Operations on Radical Expressions (Lumen Learning) Radicals (Course Hero) Operations on Radical Expressions (Lumen Learning) Radical Laws (Solitary Road) Let’s practice more by answering the exercise below. Exercise 2: Adding and Subtracting Radical Expressions Introduction: Simplify the following expressions by adding/ subtracting the radicals. 3 − 7√72 3 4 √6 − 7√54 + 5 √9  3 + 6√28 − 5 √16 3+ √32 3− 2√63 8 √2 3 2 √32 + 8 √40 3− 2√50 + 3√5 Click here to check your answer 31 PART VII: EQUATIONS WITH RADICALS Video on solving Radical Equations There are equations where radicals are present in which the variable is part of the radicands. These expressions are called radical equations or irrational equations. Examples of these are: = √ + 6 √2 − 3 = 1 – 3 3 = √ + 5 4 –2=0 – 4 + √ − 4 = 0 2 + = √5 + 6 √ + 4 At first glance, these equations may look difficult to solve, but once you get familiar with the technique, solving these would be easy. = √ + 6 4 −2=0 √ + 4 Isolate the radical to = √ + 6 4 =2 one side. √ + 4 Raise both sides of ( )2 = (√ + 6)2 )4= (2)4 4 the equation to the 2 = + 6 (√ + 4 index of the radical. + 4 = 16 Solve for the variable 2 − − 6 = 0 = 16 − 4 in the resulting 1 2 + (−1 ) + = 12 expression (−6) = 0 =−(−1) ± √(−1)2 − 4(1)(−6) 2(1) =1 ± √1 − (−24) 2 =1 ± √25 2 =1 ± 5 2 −4 1 = 2 6 1 = 2 1 = −2 1 = 3 32 Check if the found 1 = 3 = 12 values of the 2 = −2 4 −2=0 variables make the = √ + 6 √ + 4 corresponding = √ + 6 4 −2=0 −2 = √−2 + 6 radical equations 3 = √3 + 6 √(12) + 4 true. −2 = √4 √16 4− 2 = 0 3 = √9 2−2=0 We say that a root is −2 = ±4 0=0 extraneous if it uses 3=3 the secondary −2 = −2 12 is a root. (negative) root instead of the 3 is a principal root. primary (positive) root. -2 is an extraneous root.  Let’s solve the other equations. √2 − 3 = 1 – – 4 + √ − 4 = 0 (√2 − 3)2= (1 – )2 ( – 4)2 = (−√ − 4)2 2 − 3 = 1 − 2 + 2 2 − 8 + 16 = − 4 0 = 4 − 4 + 2 2 − 9 + 20 = 0 2 − 4 + 4 = 0 ( − 2)2 = 0 1 2 + (−9 ) + 20 = 0 − 2 = 0 = 2 2 =−(−9) ± √(−9) − 4(1)(20) 2(1) Checking: √2(2) − 3 = 1 – 2 =9 ± √81 − 80 √4 − 3 = 1 – 2 2 ±√1 = −1 =9 ± √1 −1 = −1 2 =9 ± 1 2 1 =9 + 1 2 =9 − 1 2 2 10 1 = 2 8 2 = 2 1 = 5 2 = 4 Checking: 1 = 5 2 = 4 4– 4 + √4 −4 = 0 5– 4 + √5− 4 = 0 0 + √0 = 0 1 + (±√1) = 0 0=0 1 + (−1) = 0 0=0 4 is a principal root solution 5 is an extraneous root solution 33 3 2 + = √5 + 6 3 = √ + 5 (2 + )2 = (√5 + 6)2 3 )3 4 + 4 + 2 = 5 + 6 33 = (√ + 5 −2 − 1 + 2 = 0 27 = + 5 2 − − 2 = 0 27 − 5 = ( − 2)( + 1) = 0 22 = = 22 − 2 = 0 + 1 = 0 Checking: = 2 3 = −1 3 = √ + 5 3 3 = √4 + 5 Checking: 3 1 = 2 3 = √22 + 5 1 = 2 3 = √27 3 2 + = √5( ) + 6 2 + = √5( ) + 6 2 + (−1) = √5(−1) + 6 2 + 2 = √5(2)+ 6 1 = √−5 + 6 4 = √10 + 6 1√1 4 = √16 1=1 4=4 Both 4 and 1 are solutions.  (4 − 8)14 + 5 = 7 3 +8=5 √4 − 3 (4 − 8)14 = 7 – 5 3 =5–8 (4 − 8)14 = 2 √4 − 3 (4 − 3)13 = − 3 ((4 − 8)14)4= 24 [(4 − 3)13]3 = (−3)3 4 − 8 = 16 4 − 3 = −27 4 = 16 + 8 4 = − 27 + 3 4 = 24 4 = − 24 24 = 4 =− 24 = 6 4 = − 6 Checking: (4(6) − 8)14 + 5 = 7 Checking: (24 − 8)14 + 5 = 7 3 +8=5 (16)14 + 5 = 7 √4(−6) − 3 (24)14 + 5 = 7 √−27 3+ 8 = 5 2+5=7 −3 + 8 = 5 7=7 5=5 7 is a solution and principal root -6 is a solution and principal root 34  3√3 − 5 − 8 = 4 √ + 2 = √ + 16 3√3 − 5 = 4 + 8 (√ + 2)2 = (√ + 16)2 3√3 − 5 = 12 + 4√ + 4 = + 16 12 + 4√ = + 16 – 4 √3 − 5 = 3 + 4√ = + 12 √3 − 5 = 4 4√ = − + 12 (√3 − 5)2= 42 4√ = 12 3 − 5 = 16 √ = 3 3 = 16 + 5 (√ )2= 32 3 = 21 = 9 = 7 Checking: Checking: √ + 2 = √ + 16 3√3 − 5 − 8 = 4 √9 + 2 = √9 + 16 3√3(7) − 5 − 8 = 4 3 + 2 = √25 3√21 − 5 − 8 = 4 5=5 3√16 − 8 = 4 5 is a solution and principal root 3(4) − 8 = 4 12 − 8 = 4 4=4 7 is a solution and principal root The links below provide many more items involving radical equations. Try doing the items first before clicking on the explanations and answers. Solving Radical Equations (Varsity tutors) Solve Radical Equations (Open Stax) Equations with radicals (Paul’s online notes) 35 OUTPUT FOR SUBMISSION Good job in finishing the second part of this module! Now let’s it’s time to check how much you did learn. Before moving to the next module, make sure you have submitted the following in your online classroom! You may click the links bellow to download the activities. Worksheet 1: Laws or Exponents, Radicals and Simplifying Radicals Worksheet 2: Adding and subtracting radicals and solving equations with radical expressions. Module 1 IW SELF -EVALUATION Part I: Before moving to the next topic, check how much you learned by answering the following questions: a. How is an expression with fractional exponent transformed to a radical form? Give at least three examples. b. What is the difference between the rule for adding/subtracting radicals and the rule for multiplying them? c. Do you find a similarity between the rule for multiplying radicals and the rule for dividing them? If so, describe it and give at least two examples. d. How does one’s skill in factoring help in determining the root of a radicand? Give at least two examples. e. How do you describe a radical expression in its simplest form? Give at least two examples. f. How is a radical equation solved? Part II: Make a list of all the main topics, concepts, mathematical processes you encountered in this section and opposite each item, place an icon or symbol that indicates how much you understood the topic/concept/process and how much you like it. You may  use: o smiling face icon or check mark for one that you understand well or that you like or enjoy doing o question mark or doubtful face icon for one that you think there is at least one thing that still needs to be clarified about your learning of it; o an x mark, a frowning-face icon for an item that you think your understanding of it is not sufficient. It is highly advised that you review the section/s where you placed a ? and an x mark is/are discussed. You may also visit the suggested link/s where the topic/concept/process is/are explained. There are also links that provide worksheets whose items have answers or solutions. In such worksheets, for your benefit, do the items first before you click to see the answer or solution, Take time to compare your solution with the solution or answer provided. You may also do independent research on the topic. It may also help if you can personally consult someone about it like “Call a friend.” option. 36 EXERCISES/ACTIVITIES 1. Exercise 1: Simplifying expressions with Fractional Exponents 2. Exercise 2: Adding and Subtracting Radical Expressions ANSWER KEYS 1. Exercise 1: Simplifying expressions with Fractional Exponents 2. Exercise 2: Adding and Subtracting Radical Expressions REFERENCES/SOURCES: Text Orines, F., Mercado, J., Suzara, J., & Manalo, C. (2012). Next Century Mathematics. Phoenix Publishing House. Huettenmueller. R. (2021). Algebra DeMYSTiFieD 2nd Second edition byHuettenmueller (2nd Second edition). McGraw-Hill Professional. Websites 8.6 Solve Radical Equations - Intermediate Algebra 2e | OpenStax. (n.d.). Open Stax. https://openstax.org/books/intermediate-algebra-2e/pages/8-6-solve-radical equations Algebra - Equations with Radicals (Practice Problems). (n.d.). Paul’s Online Notes. https://tutorial.math.lamar.edu/problems/alg/solveradicaleqns.aspx Algebra - Radicals. (n.d.). https://tutorial.math.lamar.edu/classes/alg/Radicals.aspx Basic rules for exponentiation - Math Insight. (n.d.). Math Insight. https://mathinsight.org/exponentiation_basic_rules Lumen Learning. (n.d.-a). Operations on Radical Expressions | Beginning Algebra. https://courses.lumenlearning.com/beginalgebra/chapter/7-2-1-multiplying-and dividing-radical-expressions/ Lumen Learning. (n.d.-b). Operations on Radical Expressions | Beginning Algebra. https://courses.lumenlearning.com/beginalgebra/chapter/7-2-1-multiplying-and dividing-radical-expressions/ Pierce, Rod. (21 Jul 2020). "Squares and Square Roots in Algebra". Math Is Fun. Retrieved 14 Nov 2022 from http://www.mathsisfun.com/algebra/square-root.html Radicals. (n.d.). Course Hero. Retrieved November 14, 2022, from https://courses.lumenlearning.com/boundless-algebra/chapter/radicals / 37 Radicals. Laws. Simplification. Reduction of the index. Rationalization of the denominator. (n.d.). https://solitaryroad.com/c629.html Simplify Expressions With Rational Exponents - Precalculus. (n.d.). https://www.varsitytutors.com/precalculus-help/simplify-expressions-with-rational exponents Solving Radical Equations - Algebra II. (n.d.). Varsity Tutors. https://www.varsitytutors.com/algebra_ii-help/solving-radical-equations Squares and Square Roots in Algebra. (n.d.). Mathopolis. Retrieved November 14, 2022, from https://www.mathopolis.com/questions/q.html?id=457&t=mif&qs=457_458_1084 _ 1085_1086_2286_2287_3994_3995_3996&site=1&ref=2f616c67656272612f737 1756172652d726f6f742e68746d6c&title=5371756172657320616e64205371756 1726520526f6f747320696e20416c6765627261 Images Exponent Cliparts # 387098. (n.d.). Clipart Library. http://clipart library.com/clipart/387098.htm Exponent Cliparts #100768. (n.d.). Clipart Library. http://clipart library.com/clipart/387098.htm  Videos Khan Academy. (2010, April 16). Adding and simplifying radicals | Pre-Algebra | Khan Academy [Video]. YouTube. https://www.youtube.com/watch?v=VWlFMfPVmkU The Organic Chemistry Tutor. (2016, February 16). Simplifying Radicals With Variables, Exponents, Fractions, Cube Roots - Algebra [Video]. YouTube. https://www.youtube.com/watch?v=Llrngdh3Rrg The Organic Chemistry Tutor. (2017, February 7). Fractional Exponents [Video]. YouTube. https://www.youtube.com/watch?v=GipavLCnke0 The Organic Chemistry Tutor. (2018a, January 25). Multiplying Radical Expressions With Variables and Exponents [Video]. YouTube. https://www.youtube.com/watch?v=i0TMNeOwpSg The Organic Chemistry Tutor. (2018b, January 26). Multiplying Radical Expressions With Different Index Numbers [Video]. YouTube. https://www.youtube.com/watch?v=onOQMAcnj8U The Organic Chemistry Tutor. (2018c, January 26). Solving Radical Equations [Video]. YouTube. https://www.youtube.com/watch?v=0gicD4STzpg 38