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Radicals and Rational Exponents Copyright © Cengage Learning. All rights reserved. 9 Section 9.1 Radical Expressions Copyright © Cengage Learning. All rights reserved. Objectives 1 Simplify a perfect-square root. 2 Simplify a perfect-square root expression. 3 Simplify a perfect-cube root. 4 Simplify a perfect nth root. 3 Objectives 5 Find the domain of a square-root function and a cube-root function. 6 Use a square root to solve an application. 4 1. Simplify a perfect-square root 5 Simplify a perfect-square root When solving an equation, we often must find what number must be squared to obtain a second number a. If such a number can be found, it is called a square root of a. For example, • 0 is a square root of 0, because 02 = 0. • 4 is a square root of 16, because 42 = 16. • –4 is a square root of 16, because (–4)2 = 16. • 7xy is a square root of 49x2y2, because (7xy)2 = 49x2y2. • –7xy is a square root of 49x2y2, because (–7xy)2 = 49x2y2. All positive numbers have two real-number square roots: one that is positive and one that is negative. 6 Example 1 Find the two square roots of 121. Solution: The two square roots of 121 are 11 and –11, because 112 = 121 and (–11)2 = 121 7 Simplify a perfect-square root To express square roots, we use the symbol radical sign. For example, called a Read as “The positive square root of 121 is 11.” Read as “The negative square root of 121 is –11.” The number under the radical sign is called a radicand. 8 Simplify a perfect-square root Comment The principal square root of a positive number is always positive. Although 5 and –5 are both square roots of 25, only 5 is the principal square root. The radical expression represents 5. The radical expression – represents –5. 9 Simplify a perfect-square root Square Root of a If a 0, is the positive number, the principal square root of a, whose square is a. In symbols, If a = 0, If a 0, . The principal square root of 0 is 0. is not a real number. 10 Simplify a perfect-square root Because of the previous definition, the square root of any number squared is that number. For example, 11 Simplify a perfect-square root Numbers such as 1, 4, 9, 16, 49, and 1,600 are called integer squares, because each one is the square of an integer. The square root of every integer square is an integer. 12 Simplify a perfect-square root The square roots of many positive integers are not rational numbers. For example, is an irrational number. To find an approximate value of with a calculator, we enter these numbers and press these keys. 11 ( ) Using a scientific calculator ( ) 11 Using a graphing calculator Either way, we will see that 3.31662479 13 Simplify a perfect-square root Square roots of negative numbers are not real numbers. For example, is not a real number, because no real number squared equals –9. Square roots of negative numbers come from a set called imaginary numbers. 14 2. Simplify a perfect-square root expression 15 Simplify a perfect-square root expression If x 0, the positive number x2 has x and –x for its two square roots. To denote the principal square root of x2, we must know whether x is positive or negative. If x 0, we can write represents the positive square root of x2, which is x. If x is negative, then –x 0, and we can write represents the positive square root of x2, which is –x. 16 Simplify a perfect-square root expression If we do not know whether x is positive or negative, we must use absolute value symbols to guarantee that is positive. Definition of If x can be any real number, then 17 Example 3 Simplify each expression. Assume that x can be any real number. a. Write 16x2 as (4x)2. = |4x | Because 16x2 = (|4x|)2. Since x could be negative, absolute value symbols are needed. = 4|x| Since 4 is a positive constant in the product 4x, we write it outside the absolute value symbols. 18 Example 3 b. cont’d Factor x2 + 2x + 1. Because (x + 1) can be negative, absolute value symbols are needed. c. Because x4 = ( x2)2, and x2 0, no absolute value symbols are needed. 19 3. Simplify a perfect-cube root 20 Simplify a perfect-cube root The cube root of x is any number whose cube is x. For example, 4 is a cube root of 64, because 43 = 64. 3x2y is a cube root of 27x6y3, because (3x2y)3 = 27x6y3. –2y is a cube root of –8y3, because (–2y)3 = –8y3. 21 Simplify a perfect-cube root Cube Root of a The cube root of a is denoted as whose cube is a. In symbols, and is the number If a is any real number, then 22 Simplify a perfect-cube root We note that 64 has two real-number square roots, 8 and –8. However, 64 has only one real-number cube root, 4, because 4 is the only real number whose cube is 64. Since every real number has exactly one real cube root, it is unnecessary to use absolute value symbols when simplifying cube roots. 23 Example 4 Simplify each radical. a. Because 53 = 5 5 5 = 125 b. Because c. Because (–3x)3 = (–3x)(–3x)(–3x) = –27x3 d. e. (0.6xy2)3 = (0.6xy2)(0.6xy2)(0.6xy2) = 0.216x3y6 24 Simplify a perfect-cube root Comment The previous examples suggest that if a can be factored into three equal factors, any one of those factors is a cube root of a. 25 4. Simplify a perfect nth root 26 Simplify a perfect nth root Just as there are square roots and cube roots, there are fourth roots, fifth roots, sixth roots, and so on. In the radical is called the index (or order) of the radical. When the index is 2, the radical is a square root, and we usually do not write the index. 27 Simplify a perfect nth root Comment When n is an even number greater than 1 and x 0, is not a real number. For example, is not a real number, because no real number raised to the 4th power is –81. However, when n is odd, is a real number. 28 Simplify a perfect nth root When n is an odd natural number greater than 1, represents an odd root. Since every real number has only one real nth root when n is odd, we do not need to use absolute value symbols when finding odd roots. For example, because 35 = 243 because (–2x)7 = –128x7 29 Simplify a perfect nth root When n is an even natural number greater than 1, represents an even root. In this case, there will be one positive and one negative real nth root. For example, the two real sixth roots of 729 are 3 and –3, because 36 = 729 and (–3)6 = 729. 30 Simplify a perfect nth root When finding even roots, we use absolute value symbols to guarantee that the principal nth root is positive. = |–3| = 3 34 = (–3)4. We also could simplify this as follows: = |3x | = 3|x| (3|x|)6 = 729x6. The absolute value symbols guarantee that the sixth root is positive. 31 Example 5 Simplify each radical. a. b. c. because 54 = 625 because (–2)5 = –32 Read as “the fourth root of 625.” Read as “the fifth root of –32.” Read because as “the sixth root of d. because 107 = 107 ” Read as “the seventh root of 107.” 32 Simplify a perfect nth root When finding the nth root of an nth power, we can use the following rules. Definition of If n is an odd natural number greater than 1, then If n is an even natural number, then 33 Simplify a perfect nth root We summarize the possibilities for as follows: Definition for Assume n is a natural number greater than 1 and x is a real number. If x 0, then is the positive number such that If x = 0, then 34 Simplify a perfect nth root If x 0 and n is odd, then such that and n is even, then is the real number is not a real number. 35 5. Find the domain of a square-root function and a cube-root function 36 Find the domain of a square-root function and a cube-root function Since there is one principal square root for every nonnegative real number x, the equation f(x) = determines a function, called the square-root function. 37 Example 7 Consider the function f(x) = . a. Find the domain. b. Graph the function. c. Find the range. Solution: a. To find the domain, we note that x 0 in the function because the radicand must be nonnegative. Thus, the domain is the set of nonnegative real numbers. In interval notation, the domain is [0, ). 38 Example 7 – Solution cont’d b. We can create a table of values and plot points to obtain the graph shown in Figure 9-1(a). If we use a graphing calculator, we can choose window settings of [–1, 9] for x and [–2, 5] for y to see the graph shown in Figure 9-1(b). (a) (b) Figure 9-1 39 Example 7 – Solution cont’d c. From either graph, we can conclude that the range of the function is the set of nonnegative real numbers, which is the interval [0, ). The graph also confirms that the domain is the interval [0, ). 40 Find the domain of a square-root function and a cube-root function The graphs of many functions are translations or reflections of the square-root function. For example, if k > 0, • The graph of translated k units upward. is the graph of • The graph of is the graph of translated k units downward. • The graph of is the graph of translated k units to the left. 41 Find the domain of a square-root function and a cube-root function • The graph of is the graph of translated k units to the right. • The graph of is the graph of reflected about the x-axis. 42 Find the domain of a square-root function and a cube-root function The equation defines the cube-root function. From the graph shown in Figure 9-2(a), we can see that the domain and range of the function are the set of real numbers, ℝ, and in interval notation Figure 9-2(a) 43 Find the domain of a square-root function and a cube-root function Note that the graph of passes the vertical line test. Figures 9-2(b) and 9-2(c) show several translations of the cube-root function. (b) (c) Figure 9-2 44 6. Use a square root to solve an application. 45 Use a square root to solve an application In statistics, the standard deviation of a data set is a measure of how tightly the data points are grouped around the mean (average) of the data set. The smaller the value, the more tightly the data points are grouped around the mean. To see how to compute the standard deviation of a distribution we consider the distribution 4, 5, 5, 8, 13 and construct the following table. 46 Use a square root to solve an application The population standard deviation of the distribution is the positive square root of the mean of the numbers shown in column 4 of the table. 47 Use a square root to solve an application To the nearest hundredth, the standard deviation of the given distribution is 3.29. The symbol for the population standard deviation is , the lowercase Greek letter sigma. 48 Example 10 Which of the following distributions is more tightly grouped around the mean? a. 3, 5, 7, 8, 12 b. 1, 4, 6, 11 49 Example 10 – Solution We compute the standard deviation of each distribution. a. 50 Example 10 – Solution cont’d b. Since the standard deviation for the first distribution is less than the standard deviation for the second, the first distribution is more tightly grouped around the mean. 51