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Simplifying Radicals

Simplifying Radicals. Section 10-2. Objectives . Simplify radicals involving products Simplify radicals involving quotients. Index = 2 if not specified otherwise. radicand. radical. Radical expressions. Contain a radical. Perfect squares.

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Simplifying Radicals

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  1. Simplifying Radicals Section 10-2

  2. Objectives • Simplify radicals involving products • Simplify radicals involving quotients

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

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

  7. Your turn

  8. Simplify 28x7. 28x7 = 4x6 • 7x4x6 is a perfect square and a factor of 28x7. = 4x6 • 7xUse the Multiplication Property of Square Roots. = 2x3 7x Simplify 4x6.

  9. Your turn

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

  11. 7 5x • 3 8x = 21 40x2Multiply the whole numbers and use the Multiplication Property of Square Roots. = 21 4x2 • 10 4x2 is a perfect square and a factor of 40x2. = 21 4x2 • 10 Use the Multiplication Property of Square Roots. = 21 • 2x 10 Simplify 4x2. = 42x 10 Simplify. (continued) b. 7 5x • 3 8x

  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 = 1.5 • 54Substitute 54 for h. = 81 Multiply. = 9 Simplify 81. The distance you can see is 9 miles.

  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. 13 64 49 x4 13 64 49 x4 a. 13 64 49 x4 = Use the Division Property of Square Roots. = Use the Division Property of Square Roots. 13 8 = Simplify 64. b. 7 x2 = Simplify 49 and x4. Simplify each radical expression.

  17. Your turn

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

  19. b. = Use the Division Property of Square Roots. 75x5 48x 75x5 48x 25x4 16 25 • x4 16 = Divide the numerator and denominator by 3x. = Use the Multiplication Property of Square Roots. 5x2 4 25x4 16 = Simplify 25, x4, and 16. (continued)

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

  23. 11 12x3 11 12x3 11 12x3 b. 3x 3x 3x 3x = • Multiply by to make the denominator a perfect square. 33x 36x4 = Use the Multiplication Property of Square Roots. 33x 6x2 = Simplify 36x4. (continued) Simplify the radical expression.

  24. Your turn

  25. A radical is simplified when…

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