Teaching Guide Simplifying and Combining Radical Expressions

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1 Teaching Guide Simplifying and Combining Radical Epressions Preparing for Your Class Common Vocabulary Simplified form (of a square root, cube root, and fourth root) Natural-number perfect squares, cubes, and fourth powers Product Rule for Radical Epressions, Quotient Rule for Radical Epressions n th root of a product, product of n th roots n th root of a quotient, quotient of n th roots Like radical terms, like radicals Instruction Tips When you define the product and quotient rules for square roots, be sure to emphasize that these rules only hold for real numbers. If the radicands have negative numbers, then you are really working with imaginary numbers, and these rules do not hold. If students do not have good multiplication skills, it is unlikely that they will be able to see the largest perfect squares that are factors of numbers in the radicand. If this is the case, then I recommend using the prime factorization method (factor tree method) for finding perfect square factors in the radicand. Although it may seem obvious to you and I that the in 50 is multiplied by the radical, it may not be very obvious to the students. They are especially confused the first time they are confronted with the step where they simplify the square root of a perfect square that is net to a constant multiple, like this: 5. In their minds, they are wrestling between 5 several choices 5,, 5, and + 5. You will want to make sure to emphasize that the notation in radical epressions like 50 is really 50 when you encounter it the first few times. When students see epressions like 5, they invariably want to write 5 as the answer. To see why, consider our use of the word minus in the English language as a substitute for the word without : o I d like the hamburger minus the pickle. (remove the pickle) o Give me a sundae minus the nuts. (remove the nuts) So then, it is not surprising that to our students, 5 minus seems like it should be 5. To correct this notion, you need to help your students to see the parallel between radical like terms and all the other like terms we have encountered so far in algebra. If they can categorize radical addition and subtraction in the same place as polynomial like terms, fraction like terms, rational epression like terms, etc., then it is more likely that they will avoid these kinds of mistakes. Another way to correct this notion is to emphasize how the problem might be read as 5 square root of minus 1 square root of. This is probably a good place to discuss how the radical symbol fits into the order of operations. Of course, a square root can be represented with the eponent ½, but at this point, it is usually easier for students if you just eplain that a radical symbol is another grouping symbol, in the same category as parentheses, fraction bars, and absolute value symbols. Thus would be read as ( ), so you must evaluate 5 at the same time you would evaluate any other eponential epressions. Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

2 Now that students have played a little with addition of radicals, you will begin to see quite a few interesting mistakes that you did not teach them. For eample, students often make these mistakes: becomes becomes becomes 9 5 There are so many tempting epressions involving radicals. If students don t think carefully about each p roblem, they can easily be led astray. Teaching Your Class The Product Rule for Radicals Eamples: Simplify. 8 = 8 = 16 = 900 = = 6 5 = 5 = = 10 = 0 n If n a and b are real numbers, n n n ab = a b = = = = Simpl ifying Radicals 1. Look for perfect-square, perfect- cube, factors in the radicand.. Use the product rule for radicals to rewrite the radicand as a product.. Simplify the radicand. factors? a b Perfect square a b a b Check to see if there are still perfect square factors In the radicand. *Assume a > 0. This process can be generalized to any nth root. Student Activity: Factor Trees. In this activity, students practice simplifying radicals using the factor tree method (prime factorization). (RAD-1) Guided Learning Activity: Higher on the Factor Tree. This activity will give you a chance to bridge the already learned skills of simplifying with square roots to the new skill of simplifying higher-order roots. In the last two problems you will begin to introduce variables into the radicand. (RAD-1) Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

3 NOTE: Suppose all variables that follow represent positive numbers. Simplify Radical Variable Epressions: If the inde number is a factor of the eponent in the radicand, then use the power rule to rewrite the radicand. 0 ( ) ( ) = = 6 = = 5 5 Use the product rule for eponents to write the radicand in such a way that powers of the variable epressions m atch the inde or are a multiple of the inde. ( ) ( ) 15 1 = = = = y = yy = y y = y y Student Activity: Match Up on Simp lifying Radicals. This is a tricky match up, so make sure the students read the directions closely. These are all quick and easy radical simplifications. It s good for students to get some practice in before the epressions really start getting complicated. (RAD-15) Eamples involving numbers and variables: Simplify. 8 = = = = 6 = = = y 5y y 5y y 5y y 6 0a b = 5 aab = ab 5a= ab 5a= ab 5a ( ) 6 y = 16 y = 6 y 10 9 y 9 1 y ( ) = 81 y = 81 y = y The Quotient Rul e for Square Roots n n n a a If a and b are real numbers, then n = ( b 0). n b b Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

4 Eamples: Simplify = = 16 = = = = 0 = = = 5 = ( 5) = = = = 6 ( ) 16y y y y 5 = = = y y 8 ( ) 500 y y = = 100y = ( 10y ) = 10y 5 Combining Radical Epressions Student Activity: Thread of Like Terms II. In this activity we look at like terms in the variety of mathematical epressions in which we have seen them, and then bring in radical like terms too. In the last problem, we look at the more tricky radical like terms problems and relate these back to products of radicals as well. (RAD-16) Radical terms are said to be like radicals when they have the same radicand. Eamples: Simplify = 6 5 = 7 = cannot be simplified + 6 = cannot be simplified = Before you decide that two radicals are unlike, you must look to see if either one can be written in a more simplified form. It is possible that once in the simplified form, the radicals are like terms. Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

5 Eamples: Simplify = + 5 = + 5 = = 16 9 = 8 = = = = = + = + = = = = 6 = Student Activity: Paint by Radicals. Students gain some practice with simplifying epressions involving addition or subtraction of radicals. (RAD-17) Student Activity: Radical Addition and Multiplication Tables. This is a new twist on an old problem. For addition tables, students have to look for like terms. If there are not like terms in the row & column, then students use a large X to indicate that the terms cannot be combined. Moving on to multiplication, we hope for students to see that we can multiply any two radical terms but not add any two radical terms. (RAD-18) Student Activity: Tempting Radical Epressions. Students will need to think carefully about every problem on this page. All of them provide the opportunity to be tempted into an incorrect answer. If students begin to think mathematically instead of going with their gut instinct, then the activity has served its purpose. Scrambled answers are provided so that students cannot get too far off track. (RAD-0) Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

6 Student Activity Factor Trees Directions: Use a factor tree to find the prime factorization for each number. Build branches of the tree using factor pairs. When a 80 prime number is reached in one of the branches, circle it in red to indicate that it is one of the apples (prime factors) on the tree. To write the prime-factored form, collect all the apples on the tree. The factor tree shown here tells us that the prime factorization of is 5. To find 80, we find pairs of factors in 5 5 and bring them outside the radical as a single factor to get 5or 5. To make a particularly colorful and treelike factor tree below, use brown for all the branches, green for all the numbers, and red to circle and shade in the prime numbers. In the factor tree below, your trees will go in all directions to make the branches. The number 90 has been prime factored for you = 5 = 10 6 = = 96 = = 7 = = 180 = = 1 = = 10 = = Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

7 Guided Learning Activity Higher on the Factor Tree Recall that a square root is usually written without an inde, so a = a. In a square root we had to find pairs of a factor in order to simplify the radicand. Similarly, in a cube root, like a, we would need to find triples of a factor in order to simplify the radicand. And in a fourth root, a, we would need to find quadruples of a factor in order to simplify the radicand. Notice that the inde of the radical tells us how many matching factors we need to find in order to simplify. a a a One way to search for these pairs, triples, quadruples, etc. is to use factor trees to find the prime factorization for each radicand. Eample: Simplify: 16. We look for triples since the inde is three. Using a factor tree, we factor = = = Eample: Simplify: We look for quadruples since the inde is four. Using a factor tree, we factor = 5 = Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

8 Directions: Use factor trees to help simplify the following higher order roots. 1., a Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

9 Student Activity Match Up on Simplifying Radicals Directions (READ them): Simplify each radical epression. Then look at the remaining radicand and choose the letter that corresponds to the remaining radicand. For eample, for,, 5, and, you would choose B. If there is no radical left after simplification, choose E. If none of these choices are correct, choose F. Assume that all variables represent nonnegative numbers. A B C 5 D E No radical left F None of these Radical Epressionists Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

10 Student Activity Thread of Like Terms II Directions: Simplify each of the epressions that follow y y 8 y y 8. y 6 y y 6 y What is the lesson to be learned here about addition and subtraction in algebra and mathematics in general? 6. Now be a little more careful with this group of epressions. If the epression can t be simplified, just say so! ab ab y Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

11 Student Activity Paint by Radicals Directions: Simplify each epression and shade in the corresponding square in the grid below (that contains the correctly simplified version). Assume all variables represent positive numbers. The first one has been done for you. There s a surprise when you re finished! 1. Simplify 7 50 = 6 5 =. Simplify: Simplify: Simplify: Simplify: Simplify: Simplify: Simplify: 8y y 7 9. Simplify: 10. Simplify: 11. Simplify: b b Simplify: Simplify: 5b b 1. Simplify: Simplify: y y y b b 5 19b 10b b+ 10 5b 7 Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

12 Student Activity Radical Addition and Multiplication Tables Here is a simple addition table with radical inputs. Fill in the missing boes, use the inputs cannot be combined with the given operation. if Addition Table I: Now try this addition table that also involves variables! Again, use if you cannot combine the input terms. You may want to simplify some of the radicals before you add. Assume > 0. Addition Table II: Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

13 Here is a simple multiplication table with radical inputs. Perform the multiplications and simplify each result. Multiplication Table I: Now try this multiplication table that also involves variables! Make sure to simplify the result of each multiplication. Assume > 0. Multiplication Table II: i Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

14 Student Activity Tempting Radical Epressions Directions: The epressions on this page might try to tempt you away from the strict mathematical rules you have learned about radical epressions. So, work carefully to simplify each epression and think about every move you make! If an epression cannot be simplified, then say so Scrambled Answers: Cannot be simplified further Algebra Activities Instructor's Resource Binder, M. Andersen, Copyright 011, Cengage Learning

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