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Chapter 15 Roots and Radicals Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 15.1 Introduction to Radicals Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Radicands 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. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 3 Square Root If a is a positive number, then a is the positive square root of a and a is the negative square root of a. Also, 0 = 0. Note: A square root of a negative number is not a real number. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 4 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 b2 = a. In order to find a square root of a, you need a number that, when squared, equals a. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 5 Perfect Squares 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. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 6 Example Find each square root. a. 49 b. 25 5 16 4 7 c. 4 2 d. Approximate 5 5 ≈ 2.236067977 to three decimal places. 5 ≈ 2.236 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 7 Cube Root The cube root of a real number a 3 a b only if b 3 a Note: a is not restricted to non-negative numbers for cubes. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 8 Example Find each cube root. a. 3 27 3 b. 3 125 5 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 9 Finding nth Roots Other roots can be found, as well. The nth root of a is defined as n a b only if b n a 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. Martin-Gay, Developmental Mathematics, 2e 10 Example Find each root. a. 6 64 2 b. 5 32 2 c. 4 16 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 4th power gives –16. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 11 Simplifying Radicals Containing Variables Radicands might also contain variables and powers of variables. Now, if x is a negative number, like x = –2, then x2 2 = (2) = 4 = 2, not –2, our original x. To make sure that x2 simplifies to a nonnegative number, we have the following. For any real number a, a2 = |a| 2 For example, (9) = |–9| = 9. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 12 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. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 13 Example Find each root. a. 64x10 8x 5 b. c. 3 8x 6 2x 2 2 20 25a b 5ab10 3 4a 64 a 3 d. 3 9 b b Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 14