Mathematics 2 Past Paper (6.1.2) PDF

Subject Code Math 2 Intermediate Algebra Module Code 6.0 Radicals and Complex Numbers Lesson Code 6.1.2 Radical Expressions (Assessment) Time Allocation:...

Subject Code Math 2 Intermediate Algebra Module Code 6.0 Radicals and Complex Numbers Lesson Code 6.1.2 Radical Expressions (Assessment) Time Allocation: 30 Minutes Even-numbered items are graded. Odd-numbered items have answers at the end of the lesson. NAVIGATE TA: 28 Minutes* ATA**: Before proceeding to the next part of the discussion of rational exponents, work on these exercises first. A. Find the real solutions, if any, for each equation. 1. 3𝑥 3 + 81 = 0 2. 2𝑥 2 − 32 = 0 B. Mixed Practice. Simplify each radical if it is a real number. As instructed in LG 6.1.1, assume that all variables represent positive real numbers. 1. √121𝑥 90 3 𝑧 21 6. − √27𝑥 3 7 2. √𝑎14 𝑏28 7. √𝑥 4𝑎+16 3 4 3. √−64𝑥 6 8. √−81𝑥16 5 4. ±√0.25𝑥 8 9. √(𝑥 − 5)5 4 5. −√−16 10. √𝑥 2 − 8𝑥 + 16 (Hint: Factor the polynomial first) C. Write each expression in exponential form then simplify if possible. Assume that all variables represent positive real numbers. 3 4 1. (√𝑥 2 ) 3. 4√𝑥 2 𝑦 2 8 2. √2𝑎𝑏2 4. 5√𝑥 2 𝑦 D. Write each expression in radical notation then simplify if possible. 1 2 1. (81𝑥 8 )4 3. (−27)3 3 1 2. −164 4. 06 *TA – time allocation suggested by the teacher Mathematics 2 |Page 1 of 3 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  KNOT TA: 2 Minutes* ATA**: In this lesson, you learned that radical expressions are expressions that contain the radical symbol √. Here are key points about radicals covered in this LG that you should keep in mind: In the expression 𝑛√𝑥 , 𝑛 is called the index, 𝑥 is called the radicand, and √ is called the radical symbol. The 𝑛th root of a number 𝑏, where 𝑛 is a positive integer, is any number 𝑘 whose 𝑛th power is 𝑏 (𝑘 𝑛 = 𝑏). We call 𝑛 the degree of the root. A number 𝑏 is a perfect 𝑛th power if it can be expressed as 𝑘 𝑛 where 𝑘 and 𝑛 are nonnegative integers and 𝑛 > 1. 𝑛 The principal 𝑛th root of 𝑘, √𝑘 (𝑛 > 1, 𝑛 ∈ ℤ), is a positive real number if 𝑘 > 0. 𝑛 If 𝑘 < 0 and 𝑛 is odd, then √𝑘 is a negative real number. 𝑛 If 𝑘 < 0 and 𝑛 is even, then √𝑘 is NOT a real number. 𝑛 If 𝑘 = 0, then √𝑘 = 0. If 𝑘 is a positive real number and 𝑛 is even, then 𝑘 has exactly two real 𝑛th roots. If 𝑘 is any real number and 𝑛 is odd, then 𝑘 has only one real 𝑛th root. If 𝑘 is a negative real number and 𝑛 is even, then 𝑘 has no real 𝑛th root. 𝑛 √𝑥 𝑛 = 𝑥 when 𝑛 is odd. 𝑛 √𝑥 𝑛 = |𝑥| when 𝑛 is even. If 𝑛 is a positive integer greater than 1 and 𝑥 is a real number (𝑥 ≥ 0 when n is even), then 1 𝑥 𝑛 = 𝑛√𝑥. If 𝑚 is an integer, 𝑛 is a positive integer, and 𝑛√𝑥 is a real number, then 𝑚 𝑚 𝑛 𝑥 𝑛 = √𝑥 𝑚 = ( 𝑛√𝑥 ) REFERENCES Albarico, J.M. (2013). THINK Framework. (Based on Ramos, E.G and N. Apolinario (n.d.) Science LINKS. Quezon City: Rex Bookstore Inc.) Hall, B.C. & Fabricant, M. (1993). Algebra 2 with trigonometry. New Jersey: Prentice – Hall, Inc. Larson, R. (2010). Intermediate Algebra (Fifth Edition). California: Brooks/Cole, Cengage Learning. Martin-Gay, E. (2017). Intermediate Algebra (Custom Edition for Jones County Junior College). New Jersey: Pearson Learning Solutions. McKeague, Charles P. (1986). Intermediate Algebra (3rd Edition). USA: Harcourt Brace Jovanovich, Inc. *TA – time allocation suggested by the teacher Mathematics 2 |Page 2 of 3 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  ANSWERS TO ODD-NUMBERED EXERCISES A. 1. 3𝑥 3 + 81 = 0 Answer: 3𝑥 3 = −81 Subtract 81 from both sides of the equation 𝑥 3 = −27 Divide both sides of the equation by 3 𝑥 = −3 Since (−3)3 = −27. No other real number satisfies the equation. Check: 3(−3)3 = −81 3(−27) = −81 −81 = −81, 𝑡𝑟𝑢𝑒 B. 1. √121𝑥 90 = √(11𝑥 45 )2 = 11𝑥 45 3 3 3. √−64𝑥 6 = √(−4𝑥 2 )3 = −4𝑥 2 4 5. −√−16 is not a real number 7. √𝑥 4𝑎+16 = √(𝑥 2𝑎+8 )2 = 𝑥 2𝑎+8 5 9. √(𝑥 − 5)5 = 𝑥 − 5 C. 4 8 3 1. (√𝑥 2 ) = 𝑥 3 1 1 1 1 1 3. 4√𝑥 2 𝑦 2 = (𝑥 2 𝑦 2 )4 = ((𝑥𝑦)2 )4 = (𝑥𝑦)2 = 𝑥 2 𝑦 2 D. 1 4 4 1. (81𝑥 8 )4 = √81𝑥 8 = √(3𝑥 2 )4 = 3𝑥 2 2 2 3 3. (−27)3 = 3√(−27)2 = (√−27) = (−3)2 = 9 -END- Prepared by: Ms. Melodee T. Pacio Reviewed by: Ms. Divine Faith G. Almocera Position: Special Science Teacher (SST) II Position: Special Science Teacher (SST) I Campus: PSHS – Main Campus Campus: PSHS – CARAGA Region Campus *TA – time allocation suggested by the teacher Mathematics 2 |Page 3 of 3 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  Subject Code Math 2 Intermediate Algebra Module Code 6.0 Radicals and Complex Numbers Lesson Code 6.1.1 Radical Expressions (Lesson Proper) Time Allocation: 30 Minutes TARGET After this lesson, you should be able to ▪ define radical/radical expression and retell its brief history, ▪ solve for the 𝑛th root of a number, and ▪ change expressions with rational exponents to radicals and vice-versa. HOOK TA: 5 Minutes* ATA**: 5 Have you ever encountered expressions like −2√𝑥, 4𝑥 3√𝑦 , and √2 before? Expressions like these (expressions with the radical symbol √ ) are called radical expressions. The term “radical” was first used by Mathematicians in early 1600s to refer to irrational numbers. The term first appeared in John Pell’s book entitled “An Introduction to Algebra”. The radical symbol √ , on the other hand, was introduced by René Descartes in 1637. Before he introduced this symbol, Mathematicians only used the symbol √ which is said to be derived from the first letter of the Latin word “radix” (where the term “radical” originated from) which means root. It was first seen in print in 1525 in German Mathematician Christoff Rudolff’s book entitled “Die Cross”. Radical expressions are utilized in many fields of study – Physics, Biology, Engineering, Financial Management, and more. Before we proceed to an in-depth discussion of radicals and its applications, it will be helpful for us to first recall the definition of an exponent. We will be dealing with exponents a lot in this module. If 𝑛 is a positive integer, the 𝑛th power of 𝑎, 𝑎 𝑛 , is the product of 𝑛 factors each equal to 𝑎: 𝑎 𝑛 = 𝑎 ∙ 𝑎 ∙ 𝑎 ∙∙∙ 𝑎 𝑛 factors of 𝑎 𝑎 is called the base and 𝑛 the exponent of the power. Examples: Fifth power of 𝑥: 𝑥 5 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 4th power of 2: 24 = 2 ∙ 2 ∙ 2 ∙ 2 *TA – time allocation suggested by the teacher Mathematics 2 |Page 1 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  IGNITE TA: 25 Minutes* ATA**: 𝒏th ROOT OF A NUMBER Consider 𝑥 4 = 16. What value/s of 𝑥 will make this equation true? Since (−2)4 = 16 and 24 = 16, 𝑥 can be equal to −2 or 2. We refer to these two numbers as fourth roots of 16. Next, consider 𝑥 6 = 64. What value/s of 𝑥 will make this equation true? Since (−2)6 = 64 and 26 = 64, 𝑥 can be equal to −2 or 2. We refer to these two numbers as sixth roots of 64. We define the 𝑛th root of a number as follows: 𝒏th ROOT OF A NUMBER The 𝑛th root of a number 𝑏, where 𝑛 is a positive integer, is any number 𝑘 whose 𝑛th power is 𝑏. We call 𝑛 the degree of the root. 𝑘𝑛 = 𝑏 Below are more examples to consider: a) −5 and 5 are square roots (root of degree 2) of 25 since (−5)2 = 25 and 52 = 25 * To distinguish the roots, we refer to −5 as the negative square root of 25 and to 5 as the positive square root of 25. b) −2 is a cube root (root of degree 3) of −8, but 2 isn’t since (−2)3 = −8 but 23 = 8 ≠ −8 A nonzero number can have an 𝑛 number of distinct 𝑛th roots but some of these roots may not be real. For instance, aside from 2 and −2, 2𝑖 and −2𝑖 (imaginary numbers) are also fourth roots of 16. For now, however, we will only concern ourselves with real roots. Examples Find the real solutions, if any, of each equation. 1. 𝑥 2 − 121 = 0 Answer: 𝑥 2 = 121 Add 121 to both sides of the equation. 𝑥 = ±11 Since 112 = 121 and (−11)2 = 121 Check: If 𝑥 = 11, If 𝑥 = −11, (11)2 − 121 = 0 (−11)2 − 121 = 0 121 − 121 = 0 121 − 121 = 0 0 = 0, 𝑡𝑟𝑢𝑒 0 = 0, 𝑡𝑟𝑢𝑒 The solutions are 11 and −11. *TA – time allocation suggested by the teacher Mathematics 2 |Page 2 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  2. 3𝑥 3 = −24 Answer: 𝑥 3 = −8 Divide both sides of the equation by 3. 𝑥 = −2 Since (−2)3 = −8 No other real number satisfies this equation. Check: If 𝑥 = −2, 3(−2)3 = −24 3(−8) = −24 −24 = −24, 𝑡𝑟𝑢𝑒 The solution is −2. PERFECT 𝒏th POWER 25 and 121 are examples of perfect squares. As seen in the previous examples, these numbers can be expressed as the square of another positive integer: 25 = 52 and 121 = 112 27 is a perfect cube because it can be expressed as the cube of another positive integer: 27 = 33 In general, we say that a number 𝑏 is a perfect 𝑛th power if it can be expressed as 𝑘 𝑛 where 𝑘 and 𝑛 are nonnegative integers and 𝑛 > 1 (𝑘 𝑛 = 𝑏. Some sources add the restriction that 𝑘 must be greater than 1). Here are some of the perfect 𝑛th powers that you will encounter in this course. Try to memorize powers you are not familiar with. PERFECT 4TH PERFECT SQUARES PERFECT CUBES POWERS 02 = 0 72 = 49 03 = 0 04 = 0 12 = 1 82 = 64 13 = 1 14 = 1 22 = 4 92 = 81 23 = 8 24 = 16 32 = 9 102 = 100 33 = 27 34 = 81 42 = 16 112 = 121 43 = 64 44 = 256 52 = 25 122 = 144 53 = 125 54 = 625 62 = 36 132 = 169 63 = 216 64 = 1296 *TA – time allocation suggested by the teacher Mathematics 2 |Page 3 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  Since a number can have many distinct 𝑛th roots, it is convenient to define a principal 𝒏th root. PRINCIPAL 𝒏th ROOT 𝑛 The principal 𝑛th root of a real number 𝑘, written as √𝑘 , where 𝑛 is an integer greater than 1, is described as follows: 𝑛 PRINCIPAL 𝑛th ROOT ( √𝑘 ) EXAMPLE 𝑛 If 𝑘 > 0, then √𝑘 is a positive real Principal square root of 49 √49 = 7 number. 𝑛 5 If 𝑘 = 0, then √𝑘 is zero. Principal 5th root of 0 √0 = 0 𝑛 3 If 𝑘 < 0 and 𝑛 is odd, then √𝑘 is a Principal cube root of −27 √−27 = −3 negative real number. 𝑛 If 𝑘 < 0 and 𝑛 is even, then √𝑘 is NOT Principal 4th root of −16 Not a real number a real number. As mentioned in the Hook section, an expression with the symbol √ is called a radical expression. We name the parts of a radical expression as follows: index 𝑛 radical symbol √𝑘 radicand If a radical has no index written, the index is assumed to be 2. When asked to give THE 𝑛th root of a number, we give the number’s principal 𝒏th root. For instance, the square root of 4 is 2 (in symbols, √4 = 2). If we want to get the negative square root of 4, we put a negative sign before √4 (i.e. −√4 = −2). If we want both the positive and negative square roots of 4, we write ±√4 = ±2. Examples Find each root. 1. √100 Answer: √100 = 10 since 102 = 100 4 2. √81 4 Answer: √81 = 3 since (3)4 = 81 3 3. √−0.064 3 Answer: √−0.064 = −0.4 since (−0.4)3 = −0.064 4. −√25 Answer: −√25 = −5 (−5 is the negative square root of 25) 5. √−9 Answer: Recall that if the radicand is negative and the index is even, then there is no real root. √−9 is not a real number. There is no real number whose square is −9. *TA – time allocation suggested by the teacher Mathematics 2 |Page 4 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  Based on all the examples so far, you probably already have an idea of how we can tell the number of real 𝑛th roots that a real number has. 1. If 𝑘 is a positive real number and 𝑛 is even, then 𝑘 has exactly two real 𝑛th roots, 𝑛 𝑛 which are denoted by √𝑘 and − √𝑘. E.g. 9 has two real square roots, √9 = 3 and −√9 = −3 2. If 𝑘 is any real number and 𝑛 is odd, then 𝑘 has only one real 𝑛th root, which is 𝑛 denoted by √𝑘. 3 E.g. √27 = 3 (no other real cube roots) 3 √−8 = −2 (no other real cube roots) 3. If 𝑘 is a negative real number and 𝑛 is even, then 𝑘 has no real 𝑛th root. E.g. √−8 is not a real number 𝒏 EVALUATING √𝒙𝒏 The process of finding the 𝑛th root of a variable expression is similar to the process of finding the 𝑛th root of a numerical expression. When dealing with variable expressions, however, we have to 𝑛 be extra careful when the index of the radical is even. Consider √𝑥 𝑛. Some would automatically say 𝑛 that √𝑥 𝑛 = 𝑥, but this is NOT always true. Note: Let 𝑥 be a real number and 𝑛 be an integer greater than 2. 𝑛 √𝑥 𝑛 = 𝑥 when 𝑛 is odd. 𝑛 √𝑥 𝑛 = |𝑥| when 𝑛 is even. It is important to use the absolute value bars when the index is even because the variable may represent a negative value. The absolute value bars assure that we are getting the positive or principal 𝑛th root of the radicand. Examples Evaluate each radical expression 3 5 4 4 1. √33 2. √(−2)5 3. √(−2)4 4. √24 5. √4𝑥 2 𝑦 2 6. √𝑥 2 + 2𝑥 + 1 Answers: *Because the index is odd: 3 1. √33 = 3 5 2. √(−2)5 = −2 *Because the index is even: 4 3. √(−2)4 = |−2| = 2 (It is a common mistake for students to cancel out the index and the radicand’s exponent to get −2.) *TA – time allocation suggested by the teacher Mathematics 2 |Page 5 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  4 4. √24 = |2| = 2 5. √4𝑥 2 𝑦 2 = √(2𝑥𝑦)2 = |2𝑥𝑦| = 2|𝑥𝑦| (We are not sure if 𝑥 and 𝑦 are positive real numbers so we are keeping the absolute value bars) 6. √𝑥 2 + 2𝑥 + 1 = √(𝑥 + 1)2 = |𝑥 + 1| IMPORTANT: For the remainder of the course, unless otherwise stated, we will assume that all variable radicands of a radical with an even index are positive real numbers. Since this is so, we no longer have to put absolute value signs when simplifying radicals with an even index. Note that this will not necessarily be true in future math courses. This assumption is being made for simplicity of notation only. RATIONAL EXPONENTS So far, we have only been dealing with radicals and expressions with integer exponents. In this section, we will define expressions with rational exponents so that properties of integer exponents can also be applied to rational exponents. Here, you will also see how expressions with rational exponents relate to radicals. 1 Suppose that 𝑥 = 93. If we cube both sides of the equation, we get: 1 3 𝑥3 = (93 ) 3 = 93 = 91 𝑜𝑟 9 1 3 Since 𝑥 3 = 9, 𝑥 is the number whose cube is 9, or 𝑥 = √9. We also know that 𝑥 = 93. It 1 3 follows then that 93 = √9. 𝟏 DEFINITION OF 𝒙𝒏 If 𝑛 is a positive integer greater than 1 and 𝑥 is a real number (𝑥 ≥ 0 when 𝑛 is even), then 1 𝑥 𝑛 = 𝑛√𝑥 (Notice that the denominator of the rational exponent corresponds to the index of the radical. ) Examples Write each expression in radical notation then simplify if possible. 1 1. 83 1 3 Answer: 83 = √8 = 2 *TA – time allocation suggested by the teacher Mathematics 2 |Page 6 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  1 2. −92 1 Answer: −92 = −√9 = −3 1 3. (−9)2 1 Answer: (−9)2 = √−9 (not a real number) 1 4. 4𝑥 5 1 1 Answer: 4𝑥 5 = 4 5√𝑥 (Only 𝑥 is raised to 5 so 4 is not part of the radicand.) 1 5. (5𝑦)4 1 Answer: (5𝑦)4 = 4√5𝑦 2 Now what about a rational exponent with a numerator not equal to 1? Consider 83 and recall the properties of exponents: 2 1 2 2 2 1 3 3 83 = (83 ) = (√8 ) OR 83 = (82 )3 = √82 𝒎 DEFINITION OF 𝒙 𝒏 If 𝑚 is an integer, 𝑛 is a positive integer, and 𝑛√𝑥 is a real number, then 𝑚 𝑚 𝑛 𝑥 𝑛 = √𝑥 𝑚 = ( 𝑛√𝑥 ) Notice that the denominator of the rational exponent is the index of the radical and its numerator 𝑛 𝑚 is the power the radicand is raised to. It is important to note that √𝑥 𝑚 = ( 𝑛√𝑥 ) – this means that it doesn’t matter whether we take the 𝑛th root of 𝑥 first and then raise to 𝑚 or we raise the radicand to 2 𝑚 first and then take the 𝑛th root. Consider 83 : 23 3 83 = √82 = √64 = 4 2 3 2 83 = (√8) = 22 = 4 𝑚 𝑛 *Tip: In most cases, ( 𝑛√𝑥 ) is easier to solve than √𝑥 𝑚. Examples Write each expression in radical notation then simplify if possible. 3 1. 252 Answer: 3 3 3 252 = (√25) = 53 = 125 OR 252 = √253 = √15625 = 125 *TA – time allocation suggested by the teacher Mathematics 2 |Page 7 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.  3 2. −15 Answer: 3 3 3 5 5 5 −15 = −(√1) = −(1)3 = −1 OR −15 = −√13 = −√1 = −1 3. 41.5 3 3 3 Answer: 41.5 = 42 = (√4) = 23 = 8 OR 41.5 = 42 = √43 = √64 = 8 4 4. (3𝑥 + 7)5 Answer: 4 4 4 (3𝑥 + 7)5 = ( 5√3𝑥 + 7) OR (3𝑥 + 7)5 = 5√(3𝑥 + 7)4 Examples Write each expression in exponential form then simplify if possible. 3 5 𝑥2 1. 3√𝑦 4 2. −√10 3. √−10 4. √𝑥 2 𝑦 5 5. √ 32 Answers: 1 2 2 4 1 1 1 2 5 𝑥2 5 𝑥5 𝑥5 1. 𝑦3 2. −102 3. (−10)2 4. (𝑥 2 𝑦 5 )3 or 𝑥 3 𝑦 3 5. (32) = 1 = 2 (25 )5 REFERENCES Albarico, J.M. (2013). THINK Framework. (Based on Ramos, E.G and N. Apolinario (n.d.) Science LINKS. Quezon City: Rex Bookstore Inc.) Hall, B.C. & Fabricant, M. (1993). Algebra 2 with trigonometry. New Jersey: Prentice – Hall, Inc. Larson, R. (2010). Intermediate Algebra (Fifth Edition). California: Brooks/Cole, Cengage Learning. Martin-Gay, E. (2017). Intermediate Algebra (Custom Edition for Jones County Junior College). New Jersey: Pearson Learning Solutions. McKeague, Charles P. (1986). Intermediate Algebra (3rd Edition). USA: Harcourt Brace Jovanovich, Inc. -END- Prepared by: Ms. Melodee T. Pacio Reviewed by: Ms. Divine Faith G. Almocera Position: Special Science Teacher (SST) II Position: Special Science Teacher (SST) I Campus: PSHS – Main Campus Campus: PSHS – CARAGA Region Campus *TA – time allocation suggested by the teacher Mathematics 2 |Page 8 of 8 **ATA – actual time allocation spent by the student (for information purposes only) © 2020 Philippine Science High School System. All rights reserved. This document may contain proprietary information and may only be released to third parties with approval of management. Document is uncontrolled unless otherwise marked; uncontrolled documents are not subject to update notification.