PRE CALCULUS MATH 11 Substantive Assignment Resource Material. 4 3 would be read as 4 root 3[ it indicates to multiple 4times the square root of 3]

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1 c a b where a is the coefficient where b is the radicand where c is the index [ root] PRE CALCULUS MATH 11 Substantive Assignment Resource Material 4 would be read as 4 root [ it indicates to multiple 4times the square root of ] When there is no index number, it is a square root In order to simplify radicals we can use one of the following two methods. Method One: Choosing a factor from the radicand that has a perfect square root Simplify 20 In this case, the coeficient is 1, and the radicand is 20 [ This is called an entire radical when the coefficient is 1] We can rewrite 20 as , so we have 2 times the square root of 5, or 2 5 [ This is called a mixed radical when the coefficient is a number other than 1] Method Two: Expressing the radicand as a product of prime numbers The prime numbers we generally use are 2,, 5, 7, 11, 1, 17 Simplify 20 In this case, the coeficient is 1, and the radicand is 20 [ This is called an entire radical when the coefficient is 1] We can express 20 as a product of prime numbers so we have the square root of 2 times 2 times 5 Notice that we have 225 or one group of twos and the index is also a two We take the the pair of twos and multiply two by the coefficeint in front of the radical, so we will have 2 5 [ This is called a mixed radical when the coefficient is a number other than 1] 1

2 Example 1 Simplify the following radical using either method shown previously 162 Using Method Two Simplify 162 In this case, the coeficient is 1, and the radicand is 162 We can express 162 as a product of prime numbers Notice that we have 2 or two groups of threes and the index is also a two We take the the 2 pairs of threes and multiply ()() by the coefficeint in front of the radical, so we will have 2 which is 9 2 Using Method One Simplify 162 In this case, the coeficient is 1, and the radicand is 162 We can express 162 as 81 2 The square root of 81 is 9, so we have Example 2 What is the simplest mixed radical form of 5 80 mn, m0, n 0 Using Method Two Simplify 80m n 5 In this case, the coeficient is 1, and the radicand is 80m n We can ress m n as a product of prime numbers and iables 5 exp 80 var mn m mm m m n n n Notice that we have two groups of twos, two groups of m' s, and one group of n' s We take the the 2 pairs of twos, two pairs of m' s, and the one pair of n' s and multiply (2)(2) m m ( n) by the coefficeint of 1which is in front of the radical 2 So we will have 2 2m mn 5 mn, which is 4m n 5mn 5 2

3 Using Method One Simplify 80m n 5 In this case, the coeficient is 1, and the radicand is 80m n We can express 80m n mn m m n n , 4 5 So we will have the m n mn which is m n mn Example 4 What is the entire radical form of is a mixed radical. In order to change to an entire radical we need to move the 7[ coefficient] back inside the radical so the coefficient is isthe same as 57 7 or Now we have our entire radical 245 Example 5 Without the use of a calculator arrange in order from least to greatest. 4, 6, 2 7, 5 2 4, 6, 2 7, and 5 2, are all mixed radicals. In order to arrange them in order we need to change them to entire radicals. 4 is the same as 44 or isthe same as 6 or isthe same as 722 or isthe same as 255 or So in order from least to greatest we have : 2 7[ 48], 5 2[ 50], 4 [ 48], 6[ 54],

4 Example 6 Cube Roots Simplify: 5 16b Using Method One 16 b is a mixed radical. Notice now the index is a three. In order to 5 simplify we can write the radicand as factors, one of the factors should have a perfect cube root. 16b 8 b 2 b 5 16b 2 b 2 b 2b 2b 5 Using Method Two 16 b is a mixed radical. Notice now the index is a three. 5 16b 2222bbbbb 5 Notice that we have one group of three 2' s, and one group of three b' s We need groups of three[] prime numbers or var iables to be the same because it is a cube root, not a square root as in the previous examples 16b 2222 b bbbb 2b 2b 5 2 Example 7 Simplify: Now we are adding radicals. The rule is that when the radicands are the same we can add their coefficients. We have , notice the radicands are all 5. In this form we can add their coefficients. Notice that if their is no coefficient, there is a We can add the, 2, and the 1 to give us 6 for a final answer 4

5 Example 8 Simplify: Now we are adding radicals. The rule is that the radicands of the different terms must be the same. We have , notice the radicands are 27, 12, and 45. In this form we could not add their coefficients but if we simplified the radicands we may be able to add and subtract We can add the 9 and 2to give us for a final answer Example 9 Simplify: In this form we cannot add their coefficients but if we simplified the radicands we may be able to add and subtract We can subtract the 9 and the 6 because they have the same radicand Example 10 Simplify: We are multiplying radicals in this example. The rules for multiplying and dividing are the same. You multiply or divide the coefficients with coefficients and radicands with radicands. You do not mix them. In the example we are multipling 2 5 by both 5 and 6. It is sometimes best to write the products out as prime factors in the radicands rather than having to simplify later

6 Example 11 Simplify: In the example we are foiling so we have We cannot add any of the coefficients as the radicands are different Example 12 Simplify: In the example 6 5 and the radicands we are dividing so we have reduce the coefficients 6

7 Example 1 Simplify: In the example we are dividing so we have reduce the coefficients 6 and the radicands After reducing we still have a radical in the deno min ator, it 6 6 is an irrational number and we want to rationalize it Example 14 Simplify: In the example we are dividing, this time we have a binomial in the 6 deno min ator, in order to rationalize we will multiply by it ' s conjugate Notice the conjugate is the same except the sign between the terms is opposite Since all three of the coefficients can be reduced by the same number, we must reduce them

8 Example 15 Simplify: In the example we are dividing, this time we have a binomial in the 6 deno min ator, in order to rationalize we will multiply by it ' s conjugate Since not all of the coefficients can be reduced by the same number, wecannot reducethem Example 15 8

9 Example 16 Simplify First, think of the perfect Cubes Now think of what times 5 will give me one of the perfect cubes. In this case it is 25 x 5 =

10 Example 17 Simplify 16 Think of the number that you can times with 16 to get one of the perfect cubes 16 x 4 =

11 Example 18 Simplify Again, think of the perfect cubes What number that you can multiply with to give one of the perfect cubes? And that would be x 9 = Click on this Youtube link for more lessons on Radicals. 11

12 Practice Make sure you show all your work and check your answers. When you are done, proceeded to the Substantive Assignment

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