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Math 10C Real Numbers: Lesson #5 Mixed and Entire Radicals Objective: By the end of this lesson, you will be able to: Some radicals can be written in more than one way and still have the same value. These are called _________________________________. Definitions: A radical in the form n b is called an _________________________________. A radical in the form a n b is called a _________________________________. You can “pull apart” a radical (e.g. 4 15 to 4 15 ) and get the same answer. This is called the Multiplication Property of Radicals: We can use this property to change some entire radicals into mixed radicals. We call this _______________________ radicals or writing a radical in __________________ __________. Method 1: Find a Perfect Square Factor 1. Look for a factor of the radicand that is a ________________ ____________. * To make sure the radical is in simplest form, you must find the __________________ perfect square factor. 2. Write the radicand as a _______________ of the perfect square and its factor pair. 3. Break the radical apart at the sign into the product of two radicals. 4. Take the _____________ _________ of the perfect square. Leave the other radical as is. e.g. 1) Write 200 as a mixed radical in simplest form. Math 10C Real Numbers: Lesson #5 Method 2: Use Prime Factorization 1. Write the ____________ __________________________ of the radicand. 2. Group as many prime factors as you can into ___________. 3. Multiply ________ number from each of these pairs together. This is the number that goes ______________ the radical. 4. The factors that don’t group into pairs stay in the __________________. Multiply these numbers back together to get the number under the root sign. e.g. 2) Write 54 as a mixed radical using prime factorization. e.g. 3) The area of the square shown below is 72 cm2. Determine the length of one side as a mixed radical in simplest form. e.g. 4) Find the length of the missing side of the triangle. Express the answer as a mixed radical in simplest form. 45 5 We can also simplify cube roots, fourth roots, etc. by a similar process: Unless you have perfect cubes, fourth powers, etc. memorized, it is probably best to use prime factorization to simplify these. Instead of grouping prime factors into pairs, put them in groups equal to the __________. Math 10C Real Numbers: Lesson #5 e.g. 5) Write the following radicals in simplest form: a) 3 250 b) 4 144 We can also reverse the process to convert mixed radicals to entire radicals. e.g. 6) Write the following as entire radicals: a) 7 10 c) 34 5 Rewriting mixed radicals as whole radicals can help order them. e.g. 7) Without using a calculator, arrange the following radicals in order from greatest to least: 2 10 , 4 3 , 5 2 , 3 5 Check Your Understanding: Is it possible to write every entire radical as a mixed radical? Explain why or why not. Assignment: p. 218-219 #9-18, 20-22