Simplifying Square-Root Expressions | California Standards

1. Preview Warm Up California Standards Lesson Presentation

2. Warm Up Identify the perfect square in each set. 1. 45 81 27 1112. 156 99 8 25 3. 256 84 12 1000 4. 35 216 196 72 Write each number as a product of prime numbers. 5. 366. 64 7. 196 8. 24 81 25 196 256

3. California Standards Extension of 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

4. Vocabulary radical expression radicand

5. An expression that contains a radical sign is a radical expression. There are many types of radical expressions (such as square roots, cube roots, fourth roots, and so on), but in this chapter, you will study radical expressions that contain only square roots. Examples of radical expressions: The expression under a radical sign is the radicand. A radicand may contain numbers, variables, or both. It may contain one term or more than one term.

7. Remember that, indicates a nonnegative square root. When you simplify a square-root expression containing variables, you must be sure your answer is not negative. For example, you might think that but this is incorrect because you do not know if x is positive or negative. If x = 3, then In this case, If x = –3, then In this case, In both cases This is the correct simplification of

8. Additional Example 1: Simplifying Square-Root Expressions Simplify each expression. A. B. C.

9. Check It Out! Example 1 Simplify each expression. a. b.

10. Check It Out! Example 1 Simplify each expression. d. c.

12. Additional Example 2A: Using the Product Property of Square Roots Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots Simplify.

13. Since x is nonnegative, . Additional Example 2B: Using the Product Property of Square Roots Simplify. All variables represent nonnegative numbers. Product Property of Square Roots Product Property of Square Roots

14. Helpful Hint When factoring the radicand, use factors that are perfect squares. In Example 2A, you could have factored 18 as 6  3, but this contains no perfect squares.

15. Check It Out! Example 2a Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots Simplify.

16. Since y is nonnegative, . Check It Out! Example 2b Simplify. All variables represent nonnegative numbers. Product Property of Square Roots Product Property of Square Roots

17. Check It Out! Example 2c Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots Simplify.

19. Additional Example 3: Using the Quotient Property of Square Roots Simplify. All variables represent nonnegative numbers. B. A. Simplify. Quotient Property of Square Roots Quotient Property of Square Roots Simplify. Simplify.

20. Check It Out! Example 3 Simplify. All variables represent nonnegative numbers. a. b. Quotient Property of Square Roots Simplify. Quotient Property of Square Roots Simplify. Simplify.

21. Check It Out! Example 3c Simplify. All variables represent nonnegative numbers. Quotient Property of Square Roots Factor the radicand using perfect squares. Simplify.

22. Additional Example 4A: Using the Product and Quotient Properties Together Simplify. All variables represent nonnegative numbers. Product Property Quotient Property Write 108 as 36(3). Simplify.

23. Additional Example 4B: Using the Product and Quotient Properties Together Simplify. All variables represent nonnegative numbers. Quotient Property Product Property Simplify.

24. Caution! In the expression and 5 are not common factors. is completely simplified.

25. Check It Out! Example 4a Simplify. All variables represent nonnegative numbers. Quotient Property Product Property Write 20 as 4(5). Simplify.

26. Write as . Check It Out! Example 4b Simplify. All variables represent nonnegative numbers. Quotient Property Product Property Simplify.

27. Check It Out! Example 4c Simplify. All variables represent nonnegative numbers. Quotient Property Simplify.

28. Quadrangle 250 250 Additional Example 5: Application A quadrangle on a college campus is a square with sides of 250 feet. If a student takes a shortcut by walking diagonally across the quadrangle, how far does he walk? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot. The distance from one corner of the square to the opposite one is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2.

29. Additional Example 5 Continued Solve for c. Substitute 250 for a and b. Simplify. Factor 125,000 using perfect squares.

30. The distance is ft, or about 353.6 feet. Additional Example 5 Continued Use the Product Property of Square Roots. Simplify. Use a calculator and round to the nearest tenth.

31. 60 Check It Out! Example 5 A softball diamond is a square with sides of 60 feet. How long is a throw from third base to first base in softball? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot. 60 The distance from one corner of the square to the opposite one is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2.

32. Check It Out! Example 5 Continued Solve for c. Substitute 60 for a and b. Simplify. Factor 7,200 using perfect squares.

33. The distance is , or about 84.9 feet. Check It Out! Example 5 Continued Use the Product Property of Square Roots. Simplify. Use a calculator and round to the nearest tenth.

34. Lesson Quiz: Part I Simplify each expression. 1. 6 2. |x + 5| Simplify. All variables represent nonnegative numbers. 3. 4. 5. 6.

35. mi; 11.7 mi Lesson Quiz: Part II 7. Two archaeologists leave from the same campsite. One travels 10 miles due north and the other travels 6 miles due west. How far apart are the archaeologists? Give the answer as a radical expression in simplest form. Then estimate the distance to the nearest tenth of a mile.