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Simplifying Radicals

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. Simplifying Radicals. ALGEBRA 1 LESSON 11-1. Quick Check. 11-1. Simplify 28 x 7.

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Simplifying Radicals

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  1. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 Quick Check 11-1

  2. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 Quick Check 11-1

  3. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 Simplify each radical expression. 11-1

  4. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 (continued) b. 7 5x • 3 8x Quick Check 11-1

  5. Suppose you are looking out a fourth floor window 52 ft above the ground. Use the formula d = 1.5h to estimate the distance you can see to the horizon. Round your answer to the nearest mile. d = 1.5h = 1.5 • 52Substitute 52 for h. = 78 Multiply. 8.83176 Use a calculator. Simplifying Radicals ALGEBRA 1 LESSON 11-1 To the nearest mile, the distance you can see is 9 miles. Quick Check 11-1

  6. 49 x4 13 64 49 x4 13 64 a. 49 x4 13 64 = 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 Simplify each radical expression. Quick Check 11-1

  7. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 Simplify each radical expression. 11-1

  8. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 (continued) Quick Check 11-1

  9. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 Simplify each radical expression. 11-1

  10. 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. Simplifying Radicals ALGEBRA 1 LESSON 11-1 (continued) Simplify the radical expression. Quick Check 11-1

  11. 12 36 3 3 8 2 2 a5 2 a a3 3x 15x3 5 5x Simplifying Radicals ALGEBRA 1 LESSON 11-1 Simplify each radical expression. 1. 16 • 8 2. 4 144 3. 4.5. 48 11-1

  12. Simplify 4 3 + 3. 4 3 + 3 = 4 3 + 1 3 Both terms contain 3. = (4 + 1) 3 Use the Distributive Property to combine like radicals. = 5 3 Simplify. Operations with Radical Expressions ALGEBRA 1 LESSON 11-2 Quick Check 11-2

  13. Simplify 8 5 – 45. 8 5 – 45 = 8 5 + 9 • 5 9 is a perfect square and a factor of 45. = 8 5 – 9 • 5 Use the Multiplication Property of Square Roots. = 8 5 – 3 5 Simplify 9. = (8 – 3) 5 Use the Distributive Property to combine like terms. = 5 5 Simplify. Operations with Radical Expressions ALGEBRA 1 LESSON 11-2 Quick Check 11-2

  14. Simplify 5( 8 + 9). 5( 8 + 9) = 40 + 9 5 Use the Distributive Property. = 4 • 10 + 9 5 Use the Multiplication Property of Square Roots. = 2 10 + 9 5 Simplify. Operations with Radical Expressions ALGEBRA 1 LESSON 11-2 Quick Check 11-2

  15. Simplify ( 6 – 3 21)( 6 + 21). ( 6 – 3 21)( 6 + 21) = 36 + 126 – 3 126 – 3 441 Use FOIL. = 6 – 2 126 – 3(21) Combine like radicals and simplify 36 and 441. = 6 – 2 9 • 14 – 63 9 is a perfect square factor of 126. = 6 – 2 9 • 14 – 63 Use the Multiplication Property of Square Roots. = 6 – 6 14 – 63 Simplify 9. = –57 – 6 14 Simplify. Operations with Radical Expressions ALGEBRA 1 LESSON 11-2 Quick Check 11-2

  16. 8 7 – 3 Simplify . 8 7 – 3 7 + 3 7 + 3 = • Multiply the numerator and denominator by the conjugate of the denominator. 8( 7 + 3) 7 – 3 8( 7 + 3) 4 = Multiply in the denominator. = Simplify the denominator. = 2( 7 + 3) Divide 8 and 4 by the common factor 4. = 2 7 + 2 3 Simplify the expression. Operations with Radical Expressions ALGEBRA 1 LESSON 11-2 Quick Check 11-2

  17. a.x – 5 = 4 x = 9 Isolate the radical on the left side of the equation. ( x)2 = 92Square each side. Check: x – 5 = 4 81 – 5 4 Substitute 81 for x. 9 – 5 4 4 = 4 Solving Radical Equations ALGEBRA 1 LESSON 11-3 Solve each equation. Check your answers. x – 5 = 4 x = 81 11-3

  18. ( x – 5)2= 42Square each side. Check: x – 5 = 4 21– 5 = 4 Substitute 21 for x. 16 = 4 4 = 4 Solving Radical Equations ALGEBRA 1 LESSON 11-3 (continued) b.x – 5 = 4 x – 5 = 16 Solve for x. x = 21 Quick Check 11-3

  19. Solving Radical Equations ALGEBRA 1 LESSON 11-3 On a roller coaster ride, your speed in a loop depends on the height of the hill you have just come down and the radius of the loop in feet. The equation v = 8 h – 2r gives the velocity v in feet per second of a car at the top of the loop. 11-3

  20. Solve v = 8 h – 2r for h when v = 120 and r = 18. 120 = 8 h – 2(18) Substitute 120 for v and 18 for r. (15)2 = ( h – 36)2Square both sides. 120 8 8 h – 2(18) 8 = Divide each side by 8 to isolate the radical. 15 = h – 36 Simplify. Solving Radical Equations ALGEBRA 1 LESSON 11-3 (continued) The loop on a roller coaster ride has a radius of 18 ft. Your car has a velocity of 120 ft/s at the top of the loop. How high is the hill of the loop you have just come down before going into the loop? 225 = h – 36 261 = h Quick Check The hill is 261 ft high. 11-3

  21. Solve 3x – 4 = 2x + 3. ( 3x – 4)2 = ( 2x + 3)2Square both sides. Check: 3x – 4 = 2x + 3 3(7) – 4 2(7) + 3 Substitute 7 for x. 17 = 17 Solving Radical Equations ALGEBRA 1 LESSON 11-3 3x – 4 = 2x + 3 Simplify. 3x = 2x + 7 Add 4 to each side. x = 7 Subtract 2x from each side. The solution is 7. Quick Check 11-3

  22. Solve x = x + 12. (x)2 = ( x + 12)2Square both sides. Check: x = x + 12 4 4 + 12 –3 –3 + 12 4 = 4 –3 = 3 / Solving Radical Equations ALGEBRA 1 LESSON 11-3 x2 = x + 12 x2 – x – 12 = 0 Simplify. (x – 4)(x + 3) = 0 Solve the quadratic equation by factoring. (x – 4) = 0 or (x + 3) = 0 Use the Zero–Product Property. x = 4  or   x = –3 Solve for x. The solution to the original equation is 4. The value –3 is an extraneous solution. Quick Check 11-3

  23. Solve 3x + 8 = 2. 3x = –6 ( 3x)2 = (–6)2Square both sides. Check: 3x + 8 = 2 3(12) + 8 2   Substitute 12 for x. 36 + 8 2 6 + 8 = 2   x = 12 does not solve the original equation. / 3x + 8 = 2 has no solution. Solving Radical Equations ALGEBRA 1 LESSON 11-3 3x = 36 x = 12 Quick Check 11-3

  24. Solve each radical equation. 1. 7x – 3 = 4 2. 3x – 2 = x + 2 3. 2x + 7 = 5x – 8 4.x = 2x + 8 5. 3x + 4 + 5 = 3 5 7 2 Solving Radical Equations ALGEBRA 1 LESSON 11-3 2 5 4 no solution 11-3

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