Copyright © Cengage Learning. All rights reserved. Roots, Radical Expressions, and Radical Equations 8

Copyright © Cengage Learning. All rights reserved. Section 8.2 nth Roots and Radicands That Contain Variables

3 Objectives Find the cube root of a perfect cube. Use a calculator to find an approximation of a given radical to a specified decimal places. Graph a cube root function. Find the nth root of a perfect nth power. Simplify a radical expression that contains variables

4 nth Roots and Radicands That Contain Variables We will now reverse the cubing process and find cube roots of numbers. We also will find nth roots of numbers, where n is a natural number greater than 3.

5 Find the cube root of a perfect cube 1.

6 Find the cube root of a perfect cube To find the volume V of the cube shown in Figure 8-10, we multiply its length, width, and height. V = l  w  h V = 5  5  5 = 5 3 = 125 The volume is 125 cubic inches. Figure 8-10

7 Find the cube root of a perfect cube We have seen that the product 5  5  5 can be denoted by the exponential expression 5 3, where 5 is raised to the third power. Whenever we raise a number to the third power, we are cubing it, or finding its cube. This example illustrates that the formula for the volume of a cube with each side of length s is V = s 3.

8 Find the cube root of a perfect cube Here are some additional perfect cubes of numbers. The cube of 3 is 27, because 3 3 = 27. The cube of –3 is –27, because (–3 3 ) = –27. The cube of is, because The cube of 0 is 0, because 0 3 = 0.

9 Find the cube root of a perfect cube Suppose we know that the volume of the cube shown in Figure 8-11 is 216 cubic inches. To find the length of each side, we substitute 216 for V in the formula V = s 3 and solve for s. V = s = s 3 To solve for s, we must find a number whose cube is 216. Figure 8-11

10 Find the cube root of a perfect cube Since 6 is the only number, the sides of the cube are 6 inches long. The number 6 is called a cube root of 216, because 6 3 = 216. Here are additional examples of cube roots. 3 is a cube root of 27, because 3 3 = 27. –3 is a cube root of –27, because (–3) 3 = –27. is a cube root of, because 0 is a cube root of 0, because 0 3 = 0.

11 Find the cube root of a perfect cube In general, the following is true. Cube Roots The number b is a cube root of a if b 3 = a. All real numbers have one real cube root. As the previous examples show, a positive number has a positive cube root, a negative number has a negative cube root, and the cube root of 0 is 0.

12 Find the cube root of a perfect cube Cube Root Notation The cube root of a is denoted by = b if b 3 = a.

13 Example Find each cube root. a., because 2 3 = 8. b., because 7 3 = 343. c., because 12 3 = 1,728. d., because (–2) 3 = –8. e., because (–5) 3 = –125.

14 Use a calculator to find an approximation of a given radical to a specified decimal place 2.

15 Use a calculator to find an approximation of a given radical to a specified decimal place Cube roots of numbers such as 7 are hard to compute by hand. However, we can find a decimal approximation of with a calculator. To find an approximation of with a calculator, we can press these keys Either way, the result is approximately Using a scientific calculator Using a TI84 graphing calculator

16 Use a calculator to find an approximation of a given radical to a specified decimal place If your scientific calculator does not have a key, you can use the key. We will see later that. To find the value of, we press these keys. Numbers such as 8, 27, 64, 125, –8, and –125 are called integer cubes, because each one is the cube of an integer.

17 Use a calculator to find an approximation of a given radical to a specified decimal place The cube root of any integer cube is an integer, and therefore a rational number: and Cube roots of integers that are not integer cubes are irrational numbers. For example, and are irrational numbers.

18 Graph a cube root function 3.

19 Graph a cube root function Since every real number has one real-number cube root, there is a cube root function

20 Example 3 Graph: Solution: To graph this function, we substitute numbers for x, compute, plot the resulting ordered pairs, and connect them with a smooth curve, as shown in Figure 8-12(a). Figure 8-12(a)

21 Example 3 – Solution A calculator graph is shown in Figure 8-12(b). Figure 8-12(b) cont’d

22 Find the nth root of a perfect nth power 4.

23 Find the nth root of a perfect nth power Just as there are square roots and cube roots, there are also fourth roots, fifth roots, sixth roots, and so on. In general, nth Root of a The nth root of a is denoted by and if b n = a The number n is called the index of the radical. If n is an even natural number, a must be positive or 0. In the square root symbol, the unwritten index is understood to be 2.

24 Example Find each root. a., because 3 4 = 81. b., because 2 5 = 32. c., because (–2) 5 = – 32. d. is not a real number, because no real number raised to the fourth power is – 81.

25 Simplify a radical expression that contains variables 5.

26 Simplify a radical expression that contains variables When n is even and x  0, we say that the radical represents an even root. We can find even roots of many quantities that contain variables, provided that these variables represent positive numbers or 0. When n is odd, we say that the radical represents an odd root.

27 Example Assume that each variable represents a positive number or 0 and find each root. a. = x, because x 2 = x 2. b. = x 2, because (x 2 ) 2 = x 4. c. = x 2 y, because (x 2 y) 2 = x 4 y 2. d. = x 3 y 2, because (x 3 y 2 ) 4 = x 12 y 8. e. = 2p 3 q 2, because (2p 3 q 2 ) 6 = 64p 18 q 12.