1. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 8 Rational Exponents, Radicals, and Complex Numbers

2. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Rational Exponents, Radicals, and Complex Numbers 8.1Radical Expressions and Functions 8.2Rational Exponents 8.3Multiplying, Dividing, and Simplifying Radicals 8.4Adding, Subtracting, and Multiplying Radical Expressions 8.5Rationalizing Numerators and Denominators of Radical Expressions 8.6Radical Equations and Problem Solving 8.7Complex Numbers CHAPTER 8

3. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Complex Numbers 1.Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 8.7

4. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Imaginary unit: The number represented by i, where and i 2 =  1. Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.

5. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Write each imaginary number as a product of a real number and i. a. b.c. Solution a. b.c.

6. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Rewriting Imaginary Numbers To write an imaginary number in terms of the imaginary unit i, 1. Separate the radical into two factors, 2. Replace with i. 3. Simplify

7. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

8. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example Add or subtract. (  9 + 6i) + (6 – 13i) Solution We add complex numbers just like we add polynomials—by combining like terms. (  9 + 6i) + (6 – 13i) = (  9 + 6) + (6i – 13i ) = –3 – 7i

9. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Add or subtract. (  3 + 4i) – (4 – 12i) Solution We subtract complex numbers just like we subtract polynomials. (  3 + 4i) – (4 – 12i) = (  3 + 4i) + (  4 + 12i) =  7 + 16i

10. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Multiply. a. (8i)(  4i)b. (6i)(3 – 2i) Solution a. (8i)(  4i) b. (6i)(3 – 2i)

11. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued Multiply. c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i) Solution c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)

12. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Complex conjugate: The complex conjugate of a complex number a + bi is a – bi.

13. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example Divide. Write in standard form. Solution Rationalize the denominator.

14. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Divide. Write in standard form. Solution Rationalize the denominator.

15. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example Simplify. Solution = 1 Write i 40 as (i 4 ) 10. Write i 32 as (i 4 ) 8. Replace i 4 with 1.