1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 15 Roots and Radicals

2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 15.1 Introduction to Radicals

3. Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The symbol is called a radical or radical sign. The expression within or under a radical sign is the radicand. A radical expression is an expression containing a radical sign. Radicands

4. Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Square Root If a is a positive number, then is the positive square root of a and is the negative square root of a. Also, = 0. Note: A square root of a negative number is not a real number.

5. Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Square Roots The reverse operation of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. In order to find a square root of a, you need a number that, when squared, equals a.

6. Martin-Gay, Developmental Mathematics, 2e 66 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. They cannot be written as a quotient of integers. If needed, you can find a decimal approximation for these irrational numbers on a calculator. Otherwise, leave them in radical form. Perfect Squares

7. Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Find each square root. a. b. c. d. Approximate to three decimal places. ≈ 2.236 ≈ 2.236067977

8. Martin-Gay, Developmental Mathematics, 2e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The cube root of a real number a Note: a is not restricted to non-negative numbers for cubes. Cube Root

9. Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Find each cube root. a. b.

10. Martin-Gay, Developmental Mathematics, 2e 10 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. Finding nth Roots

11. Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Find each root. a. b. c. is not a real number since the index, 4, is even and the radicand, –16, is negative. There is no real number that when raised to the 4 th power gives –16.

12. Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Radicands might also contain variables and powers of variables. To make sure that simplifies to a nonnegative number, we have the following. For any real number a, Simplifying Radicals Containing Variables x2x2 Now, if x is a negative number, like x = –2, then x2x2 = = 2, not –2, our original x. = a2a2 = |a| = |–9| For example, = 9.

13. Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplifying Radicals Containing Variables To avoid this confusion, for the rest of the chapter we assume that if a variable appears in the radicand of a radical expression, it represents positive numbers only.

14. Martin-Gay, Developmental Mathematics, 2e 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Find each root. a. b. c. d. Example