1. Section 1Chapter 8

2. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 5 3 4 Radical Expressions and Graphs Find roots of numbers. Find principal roots. Graph functions defined by radical expressions. Find nth roots of nth powers. Use a calculator to find roots. 8.1

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find roots of numbers. Objective 1 Slide 8.1- 3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The opposite (or inverse) of squaring a number is taking its square root. We now extend our discussion of roots to include cube roots fourth roots and higher roots. Slide 8.1- 4 Find roots of numbers. The nth root of a, written is a number whose nth power equals a. That is,

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The number a is the radicand. n is the index or order. The expression is the radical. Index Radicand Radical sign Radical Slide 8.1- 5 Find roots of numbers.

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Slide 8.1- 6 CLASSROOM EXAMPLE 1 Simplifying Higher Roots Solution:

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find principal roots. Objective 2 Slide 8.1- 7

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. nth Root Case 1 If n is even and a is positive or 0, then Case 2 If n is even and a is negative, then Case 3 If n is odd, then there is exactly one real nth root of a, written represents the principal nth root of a, represents the negative nth root of a. is not a real number. Slide 8.1- 8 Find principal roots.

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each root. Not a real number. Slide 8.1- 9 CLASSROOM EXAMPLE 2 Finding Roots Solution:

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph functions defined by radical expressions. Objective 3 Slide 8.1- 10

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Square Root Function The domain and range of the square root function are [0,  ). Slide 8.1- 11 Graph functions defined by radical expressions.

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The domain and range of the cube function are ( ,  ). Slide 8.1- 12 Graph functions defined by radical expressions. Cube Root Function

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph the function by creating a table of values. Give the domain and range. xf(x)f(x) –2 –1 0 2 Domain: [  2,  ) Range: [0,  ) Slide 8.1- 13 CLASSROOM EXAMPLE 3 Graphing Functions Defined with Radicals Solution:

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Xf(x)f(x) 0 1 2 33 4 Domain: ( ,  ) Range: ( ,  ) Slide 8.1- 14 CLASSROOM EXAMPLE 3 Graphing Functions Defined with Radicals (cont’d) Graph the function by creating a table of values. Give the domain and range. Solution:

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find nth roots of nth powers. Objective 4 Slide 8.1- 15

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. For any real number a, That is, the principal square root of a 2 is the absolute value of a. Slide 8.1- 16 Find nth roots of nth powers.

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each square root. Slide 8.1- 17 CLASSROOM EXAMPLE 4 Simplifying Square Roots by Using Absolute Value Solution:

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. If n is an even positive integer, then If n is an odd positive integer, then That is, use absolute value when n is even; absolute value is not necessary when n is odd. Slide 8.1- 18 Find nth roots of nth powers.

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify each root. Slide 8.1- 19 CLASSROOM EXAMPLE 5 Simplifying Higher Roots by Using Absolute Value Solution:

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use a calculator to find roots. Objective 5 Slide 8.1- 20

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use a calculator to approximate each radical to three decimal places. Slide 8.1- 21 CLASSROOM EXAMPLE 6 Finding Approximations for Roots Solution:

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. In electronics, the resonant frequency f of a circuit may be found by the formula where f is the cycles per second, L is in henrys, and C is in farads. (Henrys and farads are units of measure in electronics). Find the resonant frequency f if L = 6  10 -5 and C = 4  10 -9. About 325,000 cycles per second. Slide 8.1- 22 CLASSROOM EXAMPLE 7 Using Roots to Calculate Resonant Frequency Solution: