1. Simplifying Radicals Section 10-2

2. Objectives Simplify radicals involving products Simplify radicals involving quotients

3. Radical expressions Contain a radical Index = 2 if not specified otherwise radical radicand

4. Perfect squares The way we simplify radicals is by removing perfect square factors from the radicand, such as…. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, etc

5. Multiplication Property of Square Roots We will use this property to simplify radical expressions.

6. Simplify 243. 243 = 81 381 is a perfect square and a factor of 243. = 81 3Use the Multiplication Property of Square Roots. = 9 3Simplify 81.

7. Your turn

8. Simplify 28x 7. 28x 7 = 4x 6 7x 4x 6 is a perfect square and a factor of 28x 7. = 4x 6 7xUse the Multiplication Property of Square Roots. = 2x 3 7x Simplify 4x 6.

9. Your turn

10. Simplify each radical expression. a. 12 32 12 32 = 12 32Use the Multiplication Property of Square Roots. = 384Simplify under the radical. = 64 664 is a perfect square and a factor of 384. = 64 6Use the Multiplication Property of Square Roots. = 8 6Simplify 64.

11. (continued) b. 7 5x 3 8x = 42x 10Simplify. = 21 2x 10Simplify 4x 2. = 21 4x 2 10 Use the Multiplication Property of Square Roots. = 21 4x 2 104x 2 is a perfect square and a factor of 40x 2. 7 5x 3 8x = 21 40x 2 Multiply the whole numbers and use the Multiplication Property of Square Roots.

12. Your turn

13. Suppose you are looking out a fourth floor window 54 ft above the ground. Use the formula d = 1.5h to estimate the distance you can see to the horizon. d = 1.5h The distance you can see is 9 miles. = 9Simplify 81. = 81Multiply. = 1.5 54Substitute 54 for h.

14. Your turn Suppose you are looking out a second floor window 25 ft above the ground. Find the distance you can see to the horizon. Round your answer to the nearest mile.

15. Division Property of Square Roots

16. Simplify each radical expression. =Simplify 64. 13 8 a. 13 64 b. 49 x 4 7 x 2 =Simplify 49 and x 4. =Use the Division Property of Square Roots. 13 64 13 64 =Use the Division Property of Square Roots. 49 x 4 49 x 4

17. Your turn

18. = 12Divide. 120 10 = 4 34 is a perfect square and a factor of 12. a. 120 10 Simplify each radical expression. = 4 3Use the Multiplication Property of Square Roots. = 2 3Simplify 4.

19. b. 75x 5 48x =Divide the numerator and denominator by 3x. 75x 5 48x 25x 4 16 =Use the Division Property of Square Roots. 25x 4 16 (continued) = Use the Multiplication Property of Square Roots. 25 x 4 16 = Simplify 25, x 4, and 16. 5x 2 4

20. Your turn

21. Rationalizing the denominator A process used to force a radicand in the denominator to be a perfect square by multiplying both the numerator & denominator by the same radical expression.

22. 3 7 3 7 = Multiply by to make the denominator a perfect square. Simplify each radical expression. a. 3 7 = Simplify 49. 3 7 7 = Use the Multiplication Property of Square Roots. 3 7 49

23. = Simplify 36x 4. 33x 6x 2 (continued) b. 11 12x 3 Simplify the radical expression. = Multiply by to make the denominator a perfect square. 3x 11 12x 3 11 12x 3 = Use the Multiplication Property of Square Roots. 33x 36x 4

24. Your turn

25. A radical is simplified when…