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Simplifying Radicals: Equations, Properties, and Techniques

This resource focuses on simplifying radicals through various methods, including multiplication and division properties of square roots. It provides detailed examples of removing perfect-square factors, simplifying expressions with variables, and rationalizing denominators. Each concept is illustrated with practical equations, helping students and learners grasp the fundamentals of simplifying radical expressions effectively. The assignment section encourages practice through exercises tailored to reinforce understanding of the material covered.

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Simplifying Radicals: Equations, Properties, and Techniques

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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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