1. Chapter 8 Section 1

2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluating Roots Find square roots. Decide whether a given root is rational, irrational, or not a real number. Find decimal approximations for irrational square roots. Use the Pythagorean theorem. Use the distance formula. Find cube, fourth, and other roots. 8.1 2 3 4 5 6

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

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find square roots. When squaring a number, multiply the number by itself. To find the square root of a number, find a number that when multiplied by itself, results in the given number. The number a is called a square root of the number a 2. Slide 8.1-4 Square Root A number b is a square root of a if b 2 = a.

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The symbol, is called a radical sign, always represents the positive square root (except that ). The number inside the radical sign is called the radicand, and the entire expression—radical sign and radicand—is called a radical. The positive or principal square root of a number is written with the symbol Radical Sign Radicand The symbol is used for the negative square root of a number. Slide 8.1-5 Find square roots. (cont’d)

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The statement is incorrect. It says, in part, that a positive number equals a negative number. Slide 8.1-6 Find square roots. (cont’d)

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find all square roots of 64. Solution: Slide 8.1-7 EXAMPLE 1 Finding All Square Roots of a Number

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each square root. Solution: Slide 8.1-8 EXAMPLE 2 Finding Square Roots

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the square of each radical expression. Solution: Slide 8.1-9 EXAMPLE 3 Squaring Radical Expressions

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Decide whether a given root is rational, irrational, or not a real number. Slide 8.1-10

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Deciding whether a given root is rational, irrational, or not a real number. All numbers with square roots that are rational are called perfect squares. Perfect Squares Rational Square Roots 25 144 A number that is not a perfect square has a square root that is irrational. Many square roots of integers are irrational. Not every number has a real number square root. The square of a real number can never be negative. Therefore, is not a real number. Slide 8.1-11

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Tell whether each square root is rational, irrational, or not a real number. Solution: Not all irrational numbers are square roots of integers. For example  (approx. 3.14159) is a irrational number that is not an square root of an integer. Slide 8.1-12 EXAMPLE 4 Identifying Types of Square Roots

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Find decimal approximations for irrational square roots. Slide 8.1-13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find decimal approximations for irrational square roots. Even if a number is irrational, a decimal that approximates the number can be found using a calculator. Slide 8.1-14

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find a decimal approximation for each square root. Round answers to the nearest thousandth. Solution: Slide 8.1-15 EXAMPLE 5 Approximating Irrational Square Roots

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Use the Pythagorean theorem. Slide 8.1-16

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Many applications of square roots require the use of the Pythagorean formula. If c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the two legs, then Use the Pythagorean theorem. Be careful not to make the common mistake thinking that equals Slide 8.1-17

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the length of the unknown side in each right triangle. Give any decimal approximations to the nearest thousandth. 11 8 ? Solution: Slide 8.1-18 EXAMPLE 6 Using the Pythagorean Theorem

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. A rectangle has dimensions of 5 ft by 12 ft. Find the length of its diagonal. 5 ft 12 ft Solution: Slide 8.1-19 EXAMPLE 7 Using the Pythagorean Theorem to Solve an Application

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 5 Use the distance formula. Slide 8.1-20

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the distance formula. Distance Formula The distance between the points and is Slide 8.1-21

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the distance between and Solution: Slide 8.1-22 EXAMPLE 8 Using the Distance Formula

23. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 6 Find cube, fourth, and other roots. Slide 8.1-23

24. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Finding the square root of a number is the inverse of squaring a number. In a similar way, there are inverses to finding the cube of a number or to finding the fourth or greater power of a number. The nth root of a is written Find cube, fourth, and other roots. In the number n is the index or order of the radical. Radical sign Index Radicand It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10. Slide 8.1-24

25. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each cube root. Slide 8.1-25 EXAMPLE 9 Finding Cube Roots Solution:

26. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find each root. Solution: Slide 8.1-26 EXAMPLE 10 Finding Other Roots