# Simplifying Radicals ALGEBRA 1 LESSON 11 1 Simplify

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

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 Simplify 28 x 7 = 4 x 6 • 7 x 4 x 6 is a perfect square and a factor of 28 x 7. = 4 x 6 • 7 x Use the Multiplication Property of Square Roots. = 2 x 3 7 x Simplify 4 x 6. Quick Check 11 -1

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

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 (continued) b. 7 5 x • 3 8 x = 21 40 x 2 Multiply the whole numbers and use the Multiplication Property of Square Roots. = 21 4 x 2 • 10 factor of 40 x 2. 4 x 2 is a perfect square and a = 21 4 x 2 • 10 Square Roots. Use the Multiplication Property of = 21 • 2 x Simplify 4 x 2. = 42 x 10 10 Simplify. 11 -1 Quick Check

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 Suppose you are looking out a fourth floor window 52 ft above the ground. Use the formula d = 1. 5 h to estimate the distance you can see to the horizon. Round your answer to the nearest mile. d = 1. 5 h = 1. 5 • 52 Substitute 52 for h. = 78 Multiply. 8. 83176 Use a calculator. To the nearest mile, the distance you can see is 9 miles. Quick Check 11 -1

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 Simplify each radical expression. a. 13 64 13 13 = 64 = b. 13 8 Use the Division Property of Square Roots. Simplify 64. 49 x 4 49 49 = x 4 = 7 x 2 Use the Division Property of Square Roots. Simplify 49 and x 4. 11 -1 Quick Check

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 Simplify each radical expression. a. 120 10 120 = 12 10 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. 11 -1

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 (continued) b. 75 x 5 48 x 75 x 5 = 48 x 25 x 4 16 Divide the numerator and denominator by 3 x. = 25 x 4 16 Use the Division Property of Square Roots. = 25 • x 4 16 5 x 2 = 4 Use the Multiplication Property of Square Roots. Simplify 25, x 4, and 16. 11 -1 Quick Check

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 Simplify each radical expression. a. 3 7 3 = • 7 7 7 = = 3 7 49 3 7 7 7 Multiply by to make the denominator a 7 perfect square. Use the Multiplication Property of Square Roots. Simplify 49. 11 -1

Simplifying Radicals ALGEBRA 1 LESSON 11 -1 (continued) Simplify the radical expression. b. 11 12 x 3 11 3 x = • 3 x 12 x 3 3 x perfect square. 33 x 36 x 4 Use the Multiplication Property of Square Roots. 33 x 6 x 2 Simplify 36 x 4. = = 3 x Multiply by to make the denominator a 11 -1 Quick Check

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

Operations with Radical Expressions ALGEBRA 1 LESSON 11 -2 Simplify 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. Quick Check 11 -2

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

Operations with Radical Expressions ALGEBRA 1 LESSON 11 -2 Simplify 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. Quick Check 11 -2

Operations with Radical Expressions ALGEBRA 1 LESSON 11 -2 Simplify ( 6 – 3 21)( 6 + 21) = 36 + 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. 11 -2 Quick Check

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

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

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

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 – 2 r gives the velocity v in feet per second of a car at the top of the loop. 11 -3

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? Solve v = 8 h – 2 r for h when v = 120 and r = 18. 120 = 8 h – 2(18) Substitute 120 for v and 18 for r. 120 = 8 h – 2(18) 8 15 = h – 36 (15)2 = ( h – 36)2 225 = h – 36 261 = h The hill is 261 ft high. Divide each side by 8 to isolate the radical. Simplify. Square both sides. Quick Check 11 -3

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

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

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

Solving Radical Equations ALGEBRA 1 LESSON 11 -3 Solve each radical equation. 1. 7 x – 3 = 4 25 2. 3 x – 2 = x + 2 7 3. 2 x + 7 = 5 x – 8 5 5. 3 x + 4 + 5 = 3 4. x = 2 x + 8 no solution 11 -3 4 2

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