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1.4 1 2 3 Skills You Need: Working With Radicals 5 2 1 The followers of the Greek mathematician Pythagoras discovered values that did not correspond to any of the rational numbers. As a result, a new type of number needed to be defined to represent these values. These values are called irrational numbers. One type of __ irrational number is of the form n , where n is not a perfect square. Such numbers are sometimes referred to as radicals. In this section, you will see how to use the operations of addition, subtraction, and multiplication with radicals. 1 irrational number • a number that cannot be expressed in the _a form , where a and b b are integers and b 0 Investigate How do you multiply radicals? 1. Copy and complete the table. Where necessary, use a scientific calculator to help you evaluate each expression, rounding to two decimal places. A __ __ 4 4 = j ___ ___ 81 81 = j ____ ____ 225 225 = j __ __ 5 5 = j ___ ___ 31 31 = j ___ __ 12 9 = j ___ ____ 23 121 = j B ______ 4 4 = j ________ 81 81 = j ___________ 225 225 = j ______ 5 5 = j ________ 31 31 = j _______ 12 9 = j __________ 23 121 = j 2. What do you notice about the results in each row? 3. What conclusion can you make from your observations? Explain. 4. Reflect a) Make __ a general conclusion about an equivalent expression __ for a b . b) Do you think that this will be true for any values of a and b? Justify your answer. 34 MHR • Functions 11 • Chapter 1 The number or expression under the radical sign is called the radicand. If the radicand is greater than or equal to zero and is not a perfect square, then the radical is an irrational number. An approximate value can be found using a calculator. In many situations, it is better to work with the exact value, so the radical form is kept. Use the radical form when an approximate answer is not good enough and an exact answer is needed. Sometimes entire radicals can be simplified by removing perfect square factors. The resulting expression is called a mixed radical. radicand • a number or expression under a radical sign entire radical __ • a radical in the form n , where ___ n > 0, such as 45 mixed radical • a radical in the form __ ab , where a 1 or −1 __ and b > 0, such as 35 Example 1 Change Entire Radicals to Mixed Radicals Express each radical as a mixed radical in simplest form. ___ a) 50 ___ b) 27 ____ c) 180 Solution ___ _______ a) 50 5 25 2 ___ __ 5 ( 25 )( 2 ) __ 5 52 ______ ___ b) 27 5 9__ 3 __ )( ) 5 ( 9 __ 3 Choose 25 2, not 5 10, as 25 is a perfect square factor. ___ __ __ Use ab = a b . ___ __ __ Use ab = a b . 5 3 3 ____ _______ c) 180 5 36 ___ 5 __ or )( ) 5 ( 36 __ 5 5 65 ____ __________ 180 5 9 4 5 __ __ __ 5 ( 9 )( 4 )( 5 ) __ 5 (3)(2) 5 __ 5 65 1.4 Skills You Need: Working With Radicals • MHR 35 Adding and subtracting radicals works in the same way as adding and subtracting polynomials. You can only add like terms or, in __ this case, __ like radicals. For example, the terms in the expression 2 3 5 7 do not have the same __ radical, __ so they cannot be added, but the terms in the expression 35 __ 6 5 have a common radical, so they can be added: __ __ 3 5 6 5 5 95 . Example 2 Add or Subtract Radicals Simplify. __ __ a) 9 7 3 7 __ ___ b) 4 3 2 27 __ ___ c) 5 8 3 18 ___ ___ ___ _1 3 2 d) 28 _ 63 _ 50 4 4 3 Solution __ __ __ a) 9 7 3 7 5 6 7 ______ __ ___ __ b) 4 3 2 27 5 4 3 2 9 3 __ __ __ 5 4 3 2 9 3 __ __ 5 4 3 2 3 3 __ __ 5 4 3 6 3 __ 5 2 3 __ ___ ______ ______ c) 5 8 3 18 5 5 4 2 39 2 5 5 5 5 _1 ___ _3 ___ __ __ __ __ 5 4 2 3 9 2 __ __ 5 2 2 3 3 2 __ __ 10 2 9 2 __ 19 2 _2 3 ___ ___ Simplify 27 first. First simplify both radicals. _1 ______ _3 ______ _2 _______ 4 4 3 __ __ ___ __ __ __ 3 1 2 _ _ _ 5 4 7 9 7 25 2 d) 28 63 50 5 4 7 9 7 25 2 4 36 MHR • Functions 11 • Chapter 1 4 4 4 3 __ __ __ 3 3 7 _ 1 2 7 _ 2 5 2 5 _ 4 4 3 __ __ __ 9 7 _ 10 2 2 7 _ 5 _ 4 4 3 __ __ __ __ 7 7 __ 10 2 10 7 _ _ _ 5 7 2 or 3 4 4 3 Example 3 Multiply Radicals Simplify fully. __ __ a) ( 2 3 )( 3 6 ) __ __ d) ( 3 5 )( 2 3 ) __ __ __ b) 2 3 ( 4 5 3 ) __ __ __ __ e) ( 2 2 3 3 )( 2 2 3 3 ) Solution __ __ __ __ a) ( 2 3 )( 3 6 )5 (2 3)( 3 6 ) ______ 5 63 6 ___ 5 618 ______ 5 69 2 __ 5 6 32 __ 5 18 2 Use the commutative property and the associative property. Multiply coefficients and then multiply radicands. Connections __ __ __ __ __ b) 2 3 ( 4 5 3 ) 5 2 3 (4) ( 2 3 )( 53 ) __ __ 5 8 3 10 9 __ 5 8 3 10(3) __ 5 8 3 30 __ __ c) 7 2 ( 6 8 11 ) __ __ __ Use the distributive property. Recall that 3(x + 2) = 3x + 6 by the distributive property. The same property can be applied to multiply radicals. __ c) 7 2 ( 6 8 11 ) 5 ( 7 2 )( 6 8 ) ( 7 2 )(11) ___ __ 5 4216 77 2 __ 5 (42)(4) 77 2 __ 5 168 772 __ __ __ __ __ __ d) ( 3 5 )( 2 3 )5 3 (2) 3 ( 3 ) 5(2) 5( 3 ) __ __ __ 5 23 9 10 53 5 2 3 3 10 53 __ __ __ 5 33 7 __ __ __ __ __ 2 __ 2 e) ( 2 2 3 3 )( 2 2 3 3 )5 ( 22 ) ( 33 ) 5 4(2) 9(3) 5 8 27 5 19 Simplify and collect like terms. Connections Recall the difference of squares: (a + b)(a — b) = a2 — b2. The factors in part e) have the same pattern. They are called conjugates. 1.4 Skills You Need: Working With Radicals • MHR 37 Example 4 Solve a Problem Using Radicals A square-based pyramid has a height of 9 cm. The volume of the pyramid is 1089 cm3. Find the exact side length of the square base, in simplified form. 9 cm Solution Con n e c t i o n s __ The answer 113 cm is exact. An approximate answer can be found using a calculator. To the nearest hundredth, the side length is 19.05 cm. Let x represent the side length of the base. 1 area of base height V 5 _ 3 1 x2(9) 1089 5 _ 3 1089 5 3x2 1089 x2 5 _ 3 x2 5 363 ____ x5 363 Only the positive root is needed because x is a length. ________ x5 121 3 __ x 5 11 3 __ The exact side length of the square base of the pyramid is 11 3 cm. Key Concepts __ ___ __ a b 5 ab for a 0 and b 0. An entire radical can be simplified to a mixed radical in simplest form by removing the largest perfect square from under the radical to form a mixed radical. ___ _______ For example, 50 5 25 __ 2 5 5 2 Like__ radicals __ can be__combined through addition and subtraction. For example, 37 27 5 5 7 . Radicals can be multiplied using the __distributive __ __ __ property. For__example,__42 ( 5 3 __ 3 )5__20 6 __ 12 2 and ( 2 3 )( 2 1 )5 4 __2 3 2 3 5 2 2__ 2 3 5 2 2 1 38 MHR • Functions 11 • Chapter 1 Communicate Your Understanding __ ___ C1 Marc is asked to simplify the expression 3 75 . He says that since the radical expressions are unlike, the terms cannot be combined. Is he correct? Explain why or why not. __ __ __ C2 Describe the steps needed to simplify the expression 3 ( 2 3 4 2 ). ____ 108 . She starts by prime factoring 108: C3 Ann wants to simplify the radical 108 2 2 3 3 3 Rayanne looks for the greatest perfect square that will divide into 108 to produce a whole number. Rayanne finds that this value is 36. Explain why both techniques will result in the same solution. A Practise For help with question 1, refer to the Investigate. 1. Simplify. __ __ __ a) 3( 4 5 ) ( 5 b) 3 __ 2 ) e) 2 3 ( 3 2 ) f) 6 2 ( 11 ) __ __ ( 2__7 ) c) 5 __ __ d) 5 3 ( __ 45 ) ___ For help with question 2, refer to Example 1. 2. Express each as a mixed radical in simplest form.___ a) 12 ____ ____ b) ___ 242 d) ____ 20 c) ____ 147 e) 252 f) 392 For help with questions 3 and 4, refer to Example 2. 3. Simplify. __ __ 5. Expand __ and__simplify. __ ___ a) 5 6 ( 2 ___ 3 ) ) b) 2 2 ( 4 14 __ e) 11 2 ( 5 3 ) f) 2 6 ( 2 6 ) __ ___ ( 10 __ ) c) 8 5 __ ( __3 ) d) 3 15 __ 2 6. Expand. Simplify where possible. __ a) 3( 8 __5 ) __ __ b)__ 3 ( 5 __2 4__ 3 ) c) 3 ( 6 3 ) __ __ ( 25 )___ d) 2 __ 5 4 __ e) 8__ 2 ( 2 __ 8 3__ 12 ) f) 3 3 ( 2 7 5 2 ) 7. Expand. __ Simplify __ where possible. __ a) 2 3 5 3 4 3 __ __ __ __ b) 11 5 4 5 5 5 6 5 __ __ __ c) 7 __ 27 __ 7 __ __ d) 2 2 8 5 3 2 4 __ __ __ __ 5 e) 6 ___ 42 ___ 36 2 ___ __ f) 2 10 10 4 10 5 4. Add or__subtract __ as indicated. ___ a) 8___ 2 4 8 ___ 32 ____ b) 4 18 3 50 ____ 200 ___ ___ For help with questions 5 to 7, refer to Example 3. a)( 2 __ 5 )( 2 __ 5 ) )( 2 4 ) __ b)( 2 __2 4 __ c)( 3 2 2 )( 5 5 2 ) __ __ d)( 3 2 5 )( 5 5 ) __ __ e)( 1 5 )( 1 5 ) __ __ f)( 4 3 7 ) ( 7 1 ) 8. Simplify. _1 ____ 4 ___ 4 ___ ____ 3 _ b) 2 20 80 125 4 ___ __ ___ 3 1 2 _ _ c) 8 50 _ 18 2 ____ 5 ____3 ___ ___ 1 1 2 2 _ _ d) 125 243 _ 45 _ 48 _1 ___ a) 54 150 __ c) 20 4___ 12 ____ 125 2__ 3 ___ d) 2__ 28 ___ 54 ____ 150 5__ 7 e) 5 3 72 ___ 243 ____ 8 ___ ___ f) 44 88 99 198 5 3 3 2 1.4 Skills You Need: Working With Radicals • MHR 39 B Connect and Apply C Extend 16. Simplify. For help with questions 9 to 11, refer to Example 4. 9. Find a simplified Selecting Tools Representing Problem Solving a) Connecting Reflecting 5 3 ___ 17. A square root is simplified by finding factors that appear twice, and leaving all other factors under the radical sign. Simplifying a cube root requires the factor to appear three times under the cube root sign. Any factor that does not appear three times is left under the cube root. Simplify each ___ cube root. _____ _____ c) 4 7 ___ 12 48 d) __ 4 5 3 b) 21 7 6 b) __ 10 50 ___ e) Communicating 6 2 ___ 5 14 __ c) _ 2 Reasoning and Proving expression for the area of each shape. __ __ 10 15 5 a) ___ 3 5 6 3 3 b) 3000 a) 54 d) 3 __ c) 1125 18. a) For what values of a is a a? __ 2 b) For what values of a is a a? Explain your reasoning. ________________ 10. Explain the steps you would need to take _____ to fully simplify 2880 . 19. Math Contest If 42 42 ... 42 5 16, how many 42’s are under the radical? A 4 11. A square has an area of 675 cm2. Find the length of a side in simplified radical form. 12. On a square game board made up of small squares of side length __ 2 cm, the diagonal has a length of 20 2 cm. How many squares are on this board? 13. Find the area and the 8 cm perimeter of the rectangle 6 cm shown. Express your answers in simplified radical form. _______ __ ___ 14. Why is 16 9 not equal to 16 9 ? Justify your reasoning. 15. Is the expression __ 1 3 a solution to the equation x2 2x 2 5 0? Explain. Reasoning and Proving Selecting Tools Representing Problem Solving Connecting Reflecting Communicating 40 MHR • Functions 11 • Chapter 1 B 8 C 12 D 16 20. Math Contest The roots of the equation ________ 3x 11 5 x 3 are m and n. A possible value for m n is A 9 B 0 A 126 B 64 C 1 D 5 ____ __ __ 21. Math Contest If 128 5 2 x , what is the value of x? C 98 D 256 22. Math Contest Given that f (a b) 5 f (a)f (b) and f (x) is always positive, what is the value of f (0)? Use Technology Use a TI-Nspire™ CAS Graphing Calculator to Explore Operations With Radicals 1. a) Turn on the TI-NspireTM CAS graphing calculator. • P ress c and select 8:System Info. Then, select 2:System Settings.... • U se the e key to scroll down to Auto or Approx and ensure that it is set to Auto. Continue down to OK, and press x twice. b) Press c and select 6:New Document. Tools • TI-Nspire™ CAS graphing calculator Select 3:Add Lists & Spreadsheet. c) Use the cursor keys on the NavPad to move to cell A1. Press / q to enter the square root __ symbol. Then, press 2 and ·. d) Move to cell B1 and enter 3 . e) Move to the cell above cell C1 and enter the formula 5a*b. Press ·. Note the result in cell C1, as shown. __ __ f) Enter 5 in cell A2 and 7 in cell B2. Note the result in cell C2. g) Try a few more examples of your choice. 2. You can use the CAS to help you change entire radicals to mixed radicals. a) Press c and select 1:Add Calculator. b) Press b and select 3:Algebra. Select 2:Factor. c) Type 50 and press ·. Note the result. d) Press / q to access the square root. e) Press / v to access the previous answer. Press ·. Note the result. f) Try this shortcut. Enter the square root symbol first. Then, enter the factor() command, followed by the 50. Press ·. g) Try a few more examples of your choice. Use Technology: Use a TI-Nspire™ CAS Graphing Calculator to Explore Operations With Radicals • MHR 41 3. You can check your work on addition or subtraction of radicals. __ __ a) Enter 9 7 3 7 and press ·. Note the result. b) Try__a few more, such as __ 4__ 3 2___ 3 5 8 3 18 Be sure you can explain where the last answer came from. c) Try some examples of your choice. 4. Try some multiplication of radicals. Start with the examples shown. Then, try some of your own. 5. Try some mixed operations. Start with the examples shown. Then, try some of your own. 42 MHR • Functions 11 • Chapter 1
