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Objectives The student will be able to: 1. simplify square roots, and 2. simplify radical expressions. Designed by Skip Tyler, Varina High School 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 What is a Perfect Square? It is a number that has a whole number square root. What numbers are perfect squares? 1 4 9 16 25 36 49, 64, 81, 100, 121, 144, ... 1. Simplify 147 Find a perfect square that goes into 147. 147 = 49i3 147 = 49i 3 147 7 3 The square root is simplified when there are no perfect squares left in the radicand. What are some strategies for finding the perfect squares in radicands? 2. Simplify 605 Find a perfect square that goes into 605. 121i5 121i 5 11 5 Compare and Contrast Find the square root of calculator. 972 with your 31.18 This means 31 and 0.18 Now simplify the square root of This means 18 times 3 972 18 3 Are these answers equivalent? 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… 1 x 2 x 3 x 4 x 5 x 6 x x x x x 2 x 2 x x 3 x 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. 49i x 7x 2 5. Simplify 8x 4i2x 12 2x 25 2x 25 Simplify 1. 2. 3. 4. 3x6 3x18 6 9x 18 9x 9x 36 6. Simplify 6 · 10 Multiply the radicals. 60 4i15 4i 15 2 15 7. Simplify 2 14 · 3 21 Multiply the coefficients and radicals. 6 294 6 49i6 6i 49i 6 6i7i 6 42 6 Simplify 1. 2. 3. 4. 4x . 2 3 4 4 3x 2 x 48 4 48x . . . 3 6x i 8x How do you know when a radical problem is done? 1. No perfect squares are in the radicand. Example: 8 2. There are no fractions in the radical. 1 Example: 4 3. There are no radicals in the denominator. Example: 1 5 Simplify. Whew! It simplified! 108 3 Divide the radicals. 108 3 36 6 Uh oh… There is a radical in the denominator! Simplify. 294 6 Divide the radicals. Whew! It simplified! 294 6 49 7 Uh oh… There is a radical in the denominator! Simplify. 200 5 Divide the radicals. 200 4 40 4·10 2 10 Simplify 8 2 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! Simplify 5 80 3 5 5 80 = · 3 5 5 80 · 3 5 5 · 16 3 5 4 · 3 1 20 = 3 5 7 Simplify Uh oh… There is a fraction in the radical! Since the fraction doesn’t reduce, split the radical up. 5 7 = How do I get rid of the radical in the denominator? 5 7 i 7 7 35 49 Multiply by the same square root to make the denominator a perfect square! 35 7 Simplify 25 3 i 3 3 = 25 3 = 75 25·3 = 3 9 25 3 Multiply by the same square root to make the denominator a perfect square! 5 3 = 3 3 Simplify 12 in two different ways. Describe which way you prefer and explain why. 3 2 Closure: Explain how you can tell if a radical expression is in simplified form.