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9.3 Simplifying Radicals Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a # that, when squared, equals a. 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 or -8 3. Find the square root: 121 11, -11 4. Find the square root: 21 or -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, ... 4 =2 16 =4 25 =5 100 = 10 144 = 12 Product Rule for Radicals If a and b are real numbers, ab a b a a if b b b 0 Simplifying Radicals Example Simplify the following radical expressions. 40 4 10 2 10 5 16 5 5 4 16 15 No perfect square factor, so the radical is already simplified. LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 = 4 2= 2 2 20 = 4 5= 2 5 32 = 16 2= 4 2 75 = 25 3= 5 3 40 = 4 10= 2 10 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 48 = 16 3= 4 3 80 = 16 5= 4 5 50 = 125 = 25 450 = 225 25 2= 25 2 5= 5 5 2= 15 2 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 18 = = 288 = = 75 = = 24 = = 72 = = 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 * To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals. 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 Multiply and then simplify 5 * 35 175 25 * 7 5 7 2 8 * 3 7 6 56 6 4 *14 6 * 2 14 12 14 2 5 * 4 20 20 100 20 *10 200 5 5* 5 25 5 7 7* 7 49 7 8 8* 8 64 8 2 2 2 2 x x* x x 2 x 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 To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator 56 7 8 4*2 2 2 This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 6 7 6 * 7 42 49 7 7 42 7 42 cannot be simplified, so we are finished. This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 5 10 1 * 2 2 10 2 2 This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 3 12 3 * 12 3 3 3 3 36 Reduce the fraction. 3 3 6 3 6 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 8 2 8 2 8 8 Whew! It simplified again! I hope they all are like this! 8 16 28 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