Objectives The student will be able to: 1. simplify square roots, and 2. simplify radical expressions. If x2 = y then x is a square root of y. In the expression 64 , is the radical sign and 64 is the radicand. 1. Find the square root: 64 8 2. Find the square root: 0.04 -0.2 3. Find the square root: 121 11, -11 4. Find the square root: 21 5. Find the square root: 5 9 441 25 81 6. Use a calculator to find each square root. Round the decimal answer to the nearest hundredth. 46.5 6.82, -6.82 What numbers are perfect squares? 1•1=1 2•2=4 3•3=9 4 • 4 = 16 5 • 5 = 25 6 • 6 = 36 49, 64, 81, 100, 121, 144, ... 1. Simplify 147 Find a perfect square that goes into 147. 147 49 3 147 49 147 7 3 3 2. Simplify 605 Find a perfect square that goes into 605. 121 5 121 11 5 5 Simplify 1. 2. 3. 4. 2 18 . 3 8 6 2 36 2 . . . 72 How do you simplify variables in the radical? x 7 Look at these examples and try to find the pattern… x x 2 x x 3 x x x 4 2 x x 5 2 x x x 6 3 x x 1 What is the answer to x x 7 3 x ? 7 x As a general rule, divide the exponent by two. The remainder stays in the radical. 4. Simplify 49x 2 Find a perfect square that goes into 49. 49 x 7x 2 5. Simplify 8x 25 4 2x 12 2x 2x 25 Simplify 1. 2. 3. 4. 3x6 3x18 6 9x 18 9x 9x 36 6. Simplify 6 10 Multiply the radicals. 60 4 15 4 15 2 15 7. Simplify 2 14 3 21 Multiply the coefficients and radicals. 6 294 6 49 6 6 49 67 6 6 42 6 Simplify 6 x 1. 2. 3. 4. 4x . 2 3 4 4 3x 2 x 48 4 48x . . . 3 8x How do you know when a radical problem is done? 1. No radicals can be simplified. Example: 8 2. There are no fractions in the radical. 1 Example: 4 3. There are no radicals in the denominator. Example: 1 5 8. Simplify. Whew! It simplified! 108 3 Divide the radicals. 108 3 36 6 Uh oh… There is a radical in the denominator! 8 2 9. Simplify 2 8 4 1 4 Whew! It simplified again! I hope they all are like this! 4 2 2 Uh oh… Another radical in the denominator! 10. Simplify 5 7 Uh oh… There is a fraction in the radical! Since the fraction doesn’t reduce, split the radical up. 5 7 5 7 How do I get rid of the radical in the denominator? 7 7 35 49 Multiply by the “fancy one” to make the denominator a perfect square! 35 7 Product Property of Radicals For any numbers a and b where a 0and b 0, ab a b Product Property of Radicals Examples 72 36 2 36 2 6 2 48 16 3 16 3 4 3 Examples: 1. 30a a 30 34 34 a 17 2. 30 54x y z 9x y z 6yz 4 4 6 4 5 7 3x y z 2 2 3 6 yz Examples: 3. 3 54a b 27a b 2b 3 7 3 3 7 3 3ab 2b 2 4. 3 60xy 4 y 15xy 3 2 2 y 15xy Quotient Property of Radicals For any numbers a and b where a 0and b 0, a b a b Examples: 1. 7 16 32 2. 25 7 7 4 16 32 25 32 4 2 5 5 Examples: 3. 4. 48 3 45 4 48 3 16 4 45 45 3 5 2 2 4 Simplest Radical Form •No perfect nth power factors other than 1. •No fractions in the radicand. •No radicals in the denominator. Adding radicals We can only combine terms with radicals if we have like radicals 6 7 5 7 3 7 6 5 3 7 8 7 Reverse of the Distributive Property Examples: 1. 2 3 + 5 + 7 3 - 2 = 2 3 + 7 3 + 5- 2 = 9 3+3 Examples: 2. 5 6 3 24 150 = 5 6 3 4 6 25 6 = 5 6 6 6 5 6 =4 6 Multiplying radicals Distributive Property 2 4 3 3 3 2 3 4 3 6 12 Multiplying radicals - FOIL 3 5 F 24 3 O 3 2 3 4 3 I L 5 2 5 4 3 6 12 10 4 15 Examples: 1. 2 3 4 5 3 6 5 O F 2 3 3 2 3 6 5 L I 4 5 3 4 5 6 5 6 12 15 4 15 120 16 15 126 Examples: 5 4 2 7 = 5 2 2 75 2 2 7 O F 2. 5 4 2 7 1010 10 2 7 I L 2 7 10 2 7 2 7 100 20 7 20 7 4 49 100 4 7 72 Conjugates Binomials of the form a b c d anda b c d where a, b, c, d are rational numbers. Ex: 5 6 Conjugate: 56 3 2 2 Conjugate: 3 2 2 What is conjugate of 2 7 3? Answer: 2 7 3 The product of conjugates is a rational number. Therefore, we can rationalize denominator of a fraction by multiplying by its conjugate. Examples: 32 35 1. 35 35 3 3 5 3 2 3 2 5 2 2 3 5 3 7 3 10 13 7 3 22 3 25 Examples: 1 2 5 6 5 2. 6 5 6 5 6 5 12 5 10 6 2 2 5 16 13 5 31