Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 8 Rational Exponents, Radicals, and Complex Numbers

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 8.1 Radicals and Radical Functions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall To find the square root of a number a, we find a number that was squared to get a. 8 2 = 64 and (  8) 2 = 64, then both 8 and  8 are square roots of 64. We denote the nonnegative, or principal square root with the radical sign. We denote the negative square root with the negative radical sign.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Principal and Negative Square Roots If a is a nonnegative number, then is the principal or nonnegative square root of a is the negative square root of a.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify. Assume that all variables represent positive numbers. a. b. c. d. e. f. g. h.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Cube Root The cube root of a real number a is written as

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the cube roots. a. b. c. d.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Other roots can be found, as well. The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify the following expressions. a. b. c. d.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall If n is an even positive integer, then If n is an odd positive integer, then

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify. a. b. c. d. e. f. g.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example If and find each function value. a. b.c.d. Solution a. b. c. d.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Graph the square root function Solution 1. Identify the domain. 2. Evaluate the function for several values of x. 3. Plot the resulting points. 4. Connect the points with a smooth curve. Domain: all nonnegative numbers, x ≥ 0.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall x y x y (0, 0) (4, 2) (1, 1) Graph 6 2 (2, ) (6, )

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Graph the functionGraph the Solution 1. Identify the domain. 2. Evaluate the function for several values of x. 3. Plot the resulting points. 4. Connect the points with a smooth curve. Domain: all real numbers

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall x y x y (0, 0) (1, 1) 28 4 (4, ) (8, 2) (-1, -1) (-4, )(-8, -2) Graph