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§10.2 Simplifying Radicals

§10.2 Simplifying Radicals. Warm-Up. Complete each equation. 1. a 3 = a 2 • a 2. b 7 = b 6 • b 3. c 6 = c 3 • c 4. d 8 = d 4 • d Find the value of each expression. 5. 4 6. 169 7. 25 8. 49.

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§10.2 Simplifying Radicals

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  1. §10.2 Simplifying Radicals

  2. Warm-Up Complete each equation. 1.a3 = a2 • a2.b7 = b6 • b 3.c6 = c3 • c4. d8 = d4 • d Find the value of each expression. 5. 4 6. 169 7. 25 8. 49

  3. 1.a3 = a(2 + 1) = a2 • a12.b7 = b(6 + 1) = b6 • b1 3.c6 = c(3 + 3) = c3 • c34.d8 = d(4 + 4) = d4 • d4 5. 4 = 2 6. 169 = 13 7. 25 = 5 8. 49 = 7 Solutions

  4. Property Multiplication Property of Square Roots For every number a > 0 and b > 0, √ab = √a • √b. Example √45 = √9 • √5 = 3 • √5 = 3√5 Property

  5. 243 = 81 • 3 81 is a perfect square and a factor of 243. = 81 • 3Use the Multiplication Property of Square Roots. = 9 3Simplify 81. Simplify 243. Example 1: Removing Perfect-Square Factors

  6. 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. Example 2: Removing Variable Factors

  7. a.12 • 32 = 384 Simplify under the radical. = 64 • 6 64 is a perfect square and a factor of 384. 12 • 32 = 12 • 32 Use the Multiplication Property of Square Roots. = 64 • 6Use the Multiplication Property of Square Roots. = 8 6Simplify 64. Example 3: Multiplying Two Radical Expressions Simplify each radical expression.

  8. 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. = 81Multiply. = 9 Simplify 81. Example 4: Using a Radical Expression The distance you can see is 9 miles.

  9. Property Division Property of Square Roots For every number a > 0 and b > 0, √a/b = √a_ √b Example √9/64 = √9 = 3 √64 8 Property

  10. 13_ 64 13 64 = Use the Division Property of Square Roots. 13 64 a. = Simplify 64. 13 8 Example 5: Simplifying Fractions Within Radicals Simplify each radical expression.

  11. a. 7 7 3 7 3 7 7 7 = • Multiply by to make the denominator a perfect square. 3 7 3 7 7 = Simplify 49. 3 7 49 = Use the Multiplication Property of Square Roots. Example 6: Rationalizing a Denominator Simplify each radical expression.

  12. Simplifying Radicals Summary Simplest Radical Form A radical expression is in simplest radical form when all three of the following statements are true. 1.) The radicand has no perfect-square factors other than 1. 2.) The radicand has no fractions. 3.) The denominator of a fraction has no radical. Summary

  13. Assignment: Pg. 623 10-44 Left

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