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Radicals and Complex Numbers 77 miL2872X_ch07_469-550 9:29:06 18:13pm Page 469 IA CONFIRMING PAGES Radicals and Complex Numbers 77 7.1 Definition of an nth Root 7.2 Rational Exponents 7.3 Simplifying Radical Expressions 7.4 Addition and Subtraction of Radicals 7.5 Multiplication of Radicals 7.6 Rationalization 7.7 Radical Equations 7.8 Complex Numbers In this chapter we study radical expressions. This includes operations on square roots, cube roots, fourth roots, and so on. We also revisit the Pythagorean theorem and its applications. As you work through this chapter, try to simplify the expressions or solve the equations. The following is a Sudoku puzzle. Use the clues given to fill in the boxes labeled a through n. Then fill in the remaining part of the grid so that every row, every column, and every 2 ϫ 3 box contains the digits 1 through 6. Clues a b a. 22 b. 23 27 5 100 c. 232 d. 21 c d e. Solution to 12x ϩ 10 Ϫ 1 ϭ 3 f. Solution to 131 ϩ x Ϫ 1 ϭ 5 e f g h 10 g. Ϫ24 16 h. Length of side a. a ` ` i j k 8 i. Solution to 23 x ϩ 123 ϭ 5 j. 1251/3 k. 25 ϩ 2 25 Ϫ 2 l m Q RQ R 1 Ϫ2/3 1 Ϫ1/2 l. Ϫ a64b a169b n m. 3 ϩ 2i Ϫ Ϫ4 Ϫ 3i ϩ Ϫ2 Ϫ 5i 1 2 1 2 1 2 n. 2102 Ϫ 82 469 miL2872X_ch07_469-550 9/29/06 06:54 AM Page 470 IA CONFIRMING PAGES 470 Chapter 7 Radicals and Complex Numbers Section 7.1 Definition of an nth Root Concepts 1. Definition of a Square Root 1. Definition of a Square Root The inverse operation to squaring a number is to find its square roots. For exam- 2. Definition of an nth Root ple, finding a square root of 36 is equivalent to asking, “what number when squared 3. Roots of Variable Expressions equals 36?” One obvious answer to this question is 6 because 6 2 ϭ 36, but Ϫ6 will also 4. Pythagorean Theorem 1 2 work, because Ϫ6 2 ϭ 36. 5. Radical Functions 1 2 Definition of a Square Root b is a square root of a if b2 ϭ a. Example 1 Identifying Square Roots TIP: All positive real numbers have two real- Identify the square roots of the real numbers. valued square roots: one Ϫ positive and one a. 25 b. 49 c. 0 d. 9 negative. Zero has only one square root, which is Solution: 0 itself. Finally, for any a. 5 is a square root of 25 because 5 2 ϭ 25. 1 2 negative real number, Ϫ5 is a square root of 25 because Ϫ5 2 ϭ 25. there are no real-valued 1 2 square roots. b. 7 is a square root of 49 because 7 2 ϭ 49. 1 2 Ϫ7 is a square root of 49 because Ϫ7 2 ϭ 49. 1 2 c. 0 is a square root of 0 because 0 2 ϭ 0. 1 2 d. There are no real numbers that when squared will equal a negative number; therefore, there are no real-valued square roots of Ϫ9. Skill Practice Identify the square roots of the real numbers. 1. 64 2. 16 3. 1 4. Ϫ100 Recall from Section 1.2 that the positive square root of a real number can be denoted with a radical sign 1 . Notation for Positive and Negative Square Roots Let a represent a positive real number. Then 1. 1a is the positive square root of a. The positive square root is also called the principal square root. 2. Ϫ1a is the negative square root of a. 3. 10 ϭ 0 Skill Practice Answers 1. Ϫ8 and 8 2. Ϫ4 and 4 3. Ϫ1 and 1 4. No real-valued square roots miL2872X_ch07_469-550 9/29/06 06:54 AM Page 471 IA CONFIRMING PAGES Section 7.1 Definition of an nth Root 471 Example 2 Simplifying a Square Root Simplify the square roots. 4 a. 136 b. c. 10.04 A 9 Solution: a. 136 denotes the positive square root of 36. 136 ϭ 6 4 4 b. denotes the positive square root of . A 9 9 4 ϭ 2 A 9 3 c. 10.04 denotes the positive square root of 0.04. 10.04 ϭ 0.2 Skill Practice Simplify the square roots. 36 5. 281 6. 7. 20.09 B49 4 The numbers 36,9 , and 0.04 are perfect squares because their square roots are rational numbers. Calculator Connections Radicals that cannot be simplified to rational numbers are irrational numbers. Recall that an irrational number cannot be written as a terminating or Use a calculator to repeating decimal. For example, the symbol 113 is used to represent the exact value approximate the values of of the square root of 13. The symbol 142 is used to represent the exact value of 113 and 142. the square root of 42. These values can be approximated by a rational number by using a calculator. 113 Ϸ 3.605551275 142 Ϸ 6.480740698 TIP: Before using a calculator to evaluate a square root, try estimating the value first. 113 must be a number between 3 and 4 because 19 6 113 6 116. 142 must be a number between 6 and 7 because 136 6 142 6 149. A negative number cannot have a real number as a square root because no real number when squared is negative. For example,1Ϫ25 is not a real number because there is no real number b for which b 2 ϭϪ25. 1 2 Skill Practice Answers 6 5. 9 6. 7. 0.3 7 miL2872X_ch07_469-550 9/29/06 06:54 AM Page 472 IA CONFIRMING PAGES 472 Chapter 7 Radicals and Complex Numbers Example 3 Evaluating Square Roots Simplify the square roots, if possible. 1 a. 1Ϫ144 b. Ϫ1144 c. 1Ϫ0.01 d. Ϫ A 9 Solution: a. 1Ϫ144 is not a real number. b. Ϫ1144 TIP: For the expression ϭϪ1 ؒ 1144 Ϫ1144, the factor of Ϫ1 is outside the radical. ϭϪ1 ؒ 12 ϭϪ12 c. 1Ϫ0.01 is not a real number. 1 d. Ϫ B9 1 ؒ ϭϪ1 B9 1 ؒ ϭϪ1 3 1 ϭϪ 3 Skill Practice Simplify the square roots, if possible. 1 8. Ϫ264 9. 2Ϫ81 10. Ϫ 11. Ϫ20.25 B 4 2. Definition of an nth Root Finding a square root of a number is the inverse process of squaring a number. This concept can be extended to finding a third root (called a cube root), a fourth root, and in general an nth root. Definition of an nth Root b is an nth root of a if bn ϭ a. For example, 2 is a fourth root of 16 because 24 ϭ 16. The radical sign 1 is used to denote the principal square root of a number. The n symbol 1 is used to denote the principal nth root of a number. In the expression n 1a, n is called the index of the radical, and a is called the radicand. For a square Skill Practice Answers 12 a 1a Ϫ root, the index is 2, but it is usually not written is denoted simply as . A rad- 8. 8 9. Not a real number 1 13 a 2 10. Not a real number 11. Ϫ0.5 ical with an index of 3 is called a cube root, denoted by . miL2872X_ch07_469-550 9/29/06 06:54 AM Page 473 IA CONFIRMING PAGES Section 7.1 Definition of an nth Root 473 n Definition of 1a n 1. If n is a positive even integer and a 7 0, then 1 a is the principal (pos- itive) nth root of a. n 2. If n 7 1 is an odd integer, then 1 a is the principal nth root of a. n 3. If n 7 1 is an integer, then 1 0 ϭ 0. For the purpose of simplifying radicals, it is helpful to know the following powers: Perfect Cubes Perfect Fourth Powers Perfect Fifth Powers 13 ϭ 1 14 ϭ 1 15 ϭ 1 23 ϭ 8 24 ϭ 16 25 ϭ 32 33 ϭ 27 34 ϭ 81 35 ϭ 243 43 ϭ 64 44 ϭ 256 45 ϭ 1024 53 ϭ 125 54 ϭ 625 55 ϭ 3125 Example 4 Identifying the Principal nth Root Calculator Connections of a Real Number A calculator can be used to Simplify the expressions, if possible. approximate nth roots by x a. 14 b. 13 64 c. 15 Ϫ32 d. 14 81 using the 1 function. On e. 16 1,000,000 f. 1Ϫ100 g. 14 Ϫ16 most calculators, the index is entered first. Solution: a. 14 ϭ 2 because 2 2 ϭ 4 1 2 b. 13 64 ϭ 4 because 4 3 ϭ 64 1 2 c. 15 Ϫ32 ϭϪ2 because Ϫ2 5 ϭϪ32 1 2 d. 14 81 ϭ 3 because 3 4 ϭ 81 1 2 e. 16 1,000,000 ϭ 10 because 10 6 ϭ 1,000,000 1 2 f. 1Ϫ100 is not a real number. No real number when squared equals Ϫ100. g. 14 Ϫ16 is not a real number. No real number when raised to the fourth power equals Ϫ16. Skill Practice Simplify, if possible. 12. 24 16 13. 23 1000 14. 25 Ϫ1 15. 25 32 16. 25 100,000 17. 2Ϫ36 18.
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