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7.1 Prime, Composite, Simplify Radicals Name _______________________________ Definitions: Prime number___________________________________________________________ Composite Number _______________________________________________________ Prime Factorization _______________________________________________________ 1. Determine the prime numbers. To prime factor a number, use the tree method or the Birthday Cake method Tree Method Birthday Cake Method 1 98 2 7 49 7 7 7 49 7 2 98 2. Prime factor each of the following: 24 _____________ 600 ____________ 108_____________ 1000 ___________ 3. What are Roots? What is the difference between 2 16 and β16 ? 2 4. βRootsβ or radicals are the inverse operation of applying exponents. You can undo a power with a radical and a radical can undo a power. For example, 32 = 9, π π β9 = 3 and 43 = 64, π π β64 = 4 3 If the index is not written, then the degree is 2. β25 = 5 β49 = 7 5. How do you find roots? A. Nice roots can be found by memory or by using a calculator. β25 β81 3 β125 4 β81 4 β4096 B. Not nice roots can be found using your calculator and rounding which gives you an approximate answer or can be simplified which gives you an exact answer. Exact Value vs Approximate Value 6. You can determine the approximate value of β75 . Determine the perfect square that is closest to but less than 75. Then determine the perfect square that is closest to but greater than 75. β16 β25 β36 β49 β64 β81 β100 β121 β144 β64 = 8 β75 = ? β81 = 9 Now you know that β75 is between 8 and 9, and you can use your calculator to calculate the approximate square root as 8.660254038 β¦ There are times when an exact solution is necessary or you need to simplify a calculation and a decimal answer makes it difficult to simplify, so we simplify the radical instead of calculating the decimal value. β75 β5 β 5 β 3 Rewrite 75 in prime factored form β5 β 5 β 3 since the index is 2 because you are finding the square root, circle any groups of 2 numbers. 5β3 is the exact simplified form of β75 because one of the numbers in the circled group is written outside the radical and the other crossed off. Any uncircled number is left under the radical symbol. 7. BEFORE YOU SIMPLIFY, CHECK WITH YOUR CALCULATOR TO SEE IF THE RADICAL IS A NICE SQUARE ROOT OR THIRD ROOT OR NTH ROOT. If you get decimals, then simplify using the following steps. Steps for Simplifying Radicals 1. Write the prime factorization of your radicand and determine the index of the radical. 2. If the index is 2, circle groups of 2 identical numbers or variables. If the index is 3, circle groups of 3 identical numbers or variables. 3. The number or variable from each circled group will show up outside the radical symbol 1 time and the rest of the circled group will be crossed off. 4. Anything left uncircled will remain under the radical. If everything under the radical symbol is circled, the radical symbol will disappear. 5. Finish simplifying by multiplying the numbers and variables outside the radical together. Do together as a class β600 Do Yourself β72 β120 β32 β125π₯ 3 π¦ 2 β36π₯ 2 π¦ 3 5β168π₯ 7 8π¦β75π¦ 5 3 4 β56π₯ 5 β144π¦ 6 8. What if there is a number in front of the radical symbol? 5β32