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11-1 Simplifying Radicals 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, ... Properties of Radicals Multiplication property of square roots Division property of square roots Properties of Radicals What does this really mean? which we all know equals 10 it can be rewritten as: or which we all know equals 10 How can I use this? To write a radical in simplest form you must make sure: • • The radicand has no perfect square factors • The radicand has no fractions The denominator of a fraction has no radical This property addresses the first point To simplify Find a perfect square that goes into 75. 2. Simplify Find a perfect square that goes into 600. 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 1 x x 4 x x 6 What is the answer to x ? x x 7 7 3 x 2 3 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. 5. Simplify 8x 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 2 15 7. Simplify Multiply the coefficients and radicals. Simplify 1. 2. 3. 4. 4x . 2 3 4 4 3x 2 x 48 4 48x . . . How do you know when a radical problem is done? 1. No radicals can be simplified. Example: not done because 4 is a factor 8 2. There are no fractions in the radical. Example: 1 not done because it is a fraction 4 3. There are no radicals in the denominator. Example: 1 not done because radical 5 is in denominator 5 Division property of square roots helps with points 2 and 3 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 Uh oh… Another radical in the denominator! Whew! It simplified again! I hope they all are like this! 2 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 How do I get rid of the radical in the denominator? 35 49 Multiply by the “fancy one” to make the denominator a perfect square! 35 7