= 4 and 4 is the principal cube root of 64.

Transcription

1 Chapter Real Numbers ad Radicals Day 1: Roots ad Radicals A2TH SWBAT evaluate radicals of ay idex Do Now: Simplify the followig: a) 2 2 b) (-2) 2 c) -2 2 d) 8 2 e) (-8) 2 f) 5 g)(-5) Based o parts a ad b, what are the possibilities for the 4? A square root of k is oe of the two equal factors whose product is k. Every positive umber has two square roots. Example: 1) x 2 = 9 2) y 2 = 5 The pricipal square root of a positive real umber is its positive square root. I geeral, whe referrig to the square root of a umber, we mea the pricipal square root. Example: 25 = The cube root of k is oe of three equal factors whose product is k. The cube root of k is writte as k. Sice = 64, 4 is oe of the three equal factors whose product is 64 ad 64 = 4. There is oly oe real umber that is the cube of 64. This real umber is called the pricipal cube root. Note: ( 4) ( 4) ( 4) = 64. Therefore, 64 = 4 ad 4 is the pricipal cube root of 64. The th root of k is oe of the equal factors whose product is k. The th root of k is writte as k. k is the radicad is the idex k is the radical If k is positive, the pricipal th root of k is the positive th root If k is egative ad is odd, the pricipal th root of k is egative If k is egative ad is eve, there is o pricipal th root of k i the set of real umbers Page 1

2 Whe a variable appears i the radicad, the radical ca be simplified if the expoet of the variable is divisible by the idex. Example: Sice 8x 6 = (2x 2 )(2x 2 )(2x 2 ), 8x 6 = 2x 2. Practice: 1) If a 2 = 169, fid all values of a. 2) Evaluate each of the followig i the set of real umbers: a. 49 b. 121b 4 c d. 81 ) Fid the legth of the loger leg of a right triagle if the measure of the shorter leg is 9 cetimeters ad the measure of the hypoteuse is 41 cetimeters. 4) State whether each of the followig represets a umber that is ratioal, irratioal, or either. a. 25 b. 8 c. 8 d. 8 Page 2

3 4 e. 0 f g. 24 h Summary The th root of k is oe of the equal factors whose product is k. The th root of k is writte as k. k is the radicad is the idex k is the radical If k is positive, the pricipal th root of k is the positive th root If k is egative ad is odd, the pricipal th root of k is egative If k is egative ad is eve, there is o pricipal th root of k i the set of real umbers Exit Ticket: For what value(s) of x is the followig radical a real umber? 9 x Homework #1: Textbook page 87 #1, 2, odd, 51 Page

4 A2TH SWBAT simplify radicals Day 2: Simplifyig Radicals Do Now: Evaluate each of the followig a) 6 b) 8 27 c) 144x 2 y 10 d) 125a 6 b 18 How would you write 12 i simplest radical form? I order to simplify a square root, we use the relatioship a b = a b We usually wat to use the greatest perfect square factor to write our aswer i simplest radical form. Simplest radical form is achieved whe the radicad is a prime umber or the product of prime umbers. Example: 72 = 6 2 = 6 2 Simplify: a) 24x 4 b) 50a c) 8x 5 y 6 d) 2 20b 17 We use the same process whe simplifyig a cube root, breakig dow the radicad usig a perfect cube factor. Example: 24 = 8 = 2 Simplify: a) 54 b) 40x 9 c) 2x 4 d) 192b 14 c 12 Fractioal Radicads: A radical is i simplest form whe the radicad is a iteger. For ay o-egative a ad positive c: a c = a c Page 4

5 Example: 2 = 2 = 6 9 = 6 9 = 6 Fractioal Radicads cotiued... Simplify: a) 8 9 b) 4 5 c) 8 d) 9a 4 8b e) f) a 2b c 2 More Practice! Page 5

6 Summary: For roots of ay idex: ab = a b ad a b = a b Exit Ticket: Simplify: a) ab5 b) 75x 5 y 6 Homework #2: Textbook page 9 #1, 2,, 7, 9, 11, 17, 19, 21, 2, 27 Page 6

7 Day : Addig, Subtractig, ad Multiplyig Radicals A2TH SWBAT add, subtract, ad multiply radicals. Do Now: Simplify: a) ab5 b) 75x 5 y 6 c) What are the rules whe addig or subtractig algebraic terms? For example, how would you simplify (b 4-5b +) - (b 4 + b + )? d) What are the rules whe multiply algebraic terms? For example, how would you simplify (a 2 b)(2abc)? We use similar rules as above whe addig, subtractig, ad multiplyig radical expressios. To express the sum or differece of two radicals as a sigle radical, the radicals must have the same idex ad the same radicad. I other words, they must be like radicals. Example: = 5 4 Simplify: a) b) 4 b + b c) Two radicals that do ot have the same radicad or do ot have the same idex are ulike radicals. Example: 2 + caot be expressed as a sigle radical Sometimes radical expressios that seem to be ulike terms ca be simplified first ad the appear as like radicals. Example: = = 7 2 Add or Subtract: a) 27b 12b b) Page 7

8 c) 8x + 16x + 27x d) x 1 x + 00x e) Solve for x: 4x 8 = 72 Recall that if a ad b are o-egative umbers, ab = a b. By the symmetric property of equality, a b = ab. We ca use this rule to multiply radicals. The Distributive Property may also be used whe eeded. Example: 8 2 = 16 = 4 Multiply: a) 6a 18a b) c) 5 x 4 8x 7 4 d) 48x 2 4 x2 e) 5 ( ) f) ( + ) (2 6 ) g) (2 + 2 )(5 2 ) h) (4 + ) ( 4 ) Page 8

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11 Summary Page 11

12 Exit Ticket: Simplify: ( 2) 2 Homework #: Textbook page 97 #1,, 7, 9, 21, 25, 29, 1,, 9; page 100 #21-45 odd Page 12

13 Day 4: Dividig Radicals ad Ratioalizig a Deomiator A2TH SWBAT 1) simplify quotiets of radical expressios ad 2) simplify radical expressios by ratioalizig the deomiator. Do Now: Simplify the followig radical expressios a) b) ( 2 7)( 4 7) Compare these two computatios: The geeral rule for the quotiet of roots: If a ad b are real umbers, with b 0, the a = a b b This rule ca also be used i the reversed order: a b = a b Example: 4 = 2 9 Example: 8 2 = 8 2 = 4 = 2 Simplify: a) 5 27 b) 8x2 49 c) 75x y x Page 1

14 A fractio is i simplest form whe the deomiator is a ratioal umber. We do ot wat a radical i the deomiator. Example 1. Ratioalize: ( 7 ) = Simplify: a) x 2 b) x c) d) 4 2 5x 4 Example 2. Ratioalize: 2+ 6 Here we use what s called the cojugate to ratioalize the deomiator. The cojugate of (2 + 6) is (2 6) ( ) = = = = Ratioalize: a) b) c) + 5 Page 14

15 d) x+ y x y e) x+y + x+y f) g) 5 x 5x h) Solve for x: x 2 = x Page 15

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17 Summary Exit Ticket: Homework #4 : Textbook page 10 #17-1 odd; page 107 #21-5 odd, odd Page 17

18 Day 5: Solvig Radical Equatios A2TH SWBAT: solve radical equatios Do Now: A radical equatio is a equatio i which the variable is hidig iside a radical sig. To solve a radical equatio, follow these steps: 1. Isolate the radical (or oe of the radicals) to oe side of the equal sig. 2. If the radical is a square root, square each side of the equatio. (If the radical is ot a square root, raise each side to a power equal to the idex of the root.). Solve the resultig equatio. 4. Check your aswer(s) to avoid extraeous roots. 1. x x = 2 Page 18

19 x = x 5x x + = x x 1 = 5x 9 7. x = 0 2x 8. x = x Page 19

20 9. 2 x + 8 = x x + 5 = x x x 5 = 2 1. x 4 = 9x 14. x x 5 = 1 Page 20

21 15. 2x + = 5 x x + 1 x = 2 Summary To solve radical equatios: 1. Isolate the radical (or oe of the radicals) to oe side of the equal sig. 2. If the radical is a square root, square each side of the equatio. (If the radical is ot a square root, raise each side to a power equal to the idex of the root.). Solve the resultig equatio. 4. Check your aswer(s) to avoid extraeous roots. Exit Ticket: Homework #5: page 112 #2, 11-5 odd Page 21

22 A2TH Packet #: Name: Teacher: Pd: Page 22

23 Table of Cotets o Day 1: SWBAT: evaluate radicals of ay idex HW: Textbook page 87 #1, 2, odd, 51 o Day 2: SWBAT: simplify radicals HW: Textbook page 9 #1, 2,, 7, 9, 11, 17, 19, 21, 2, 27 o Day : SWBAT: add, subtract ad multiply radicals HW: Textbook page 97 #1,,7,9,21,25,29,1,,9; page 100 #21 45 odd o Day 4: SWBAT: 1) simplify quotiets of radical expressios ad 2) simplify radical expressios by ratioalizig the deomiator. HW: Textbook page 10 #17-1 odd; page 107 #21-5 odd, odd o Day 5: solve radical equatios HW: page 112 #2, 11-5 odd Page 2

24 Assigmet #1 Assigmet #2 Page 24

25 Assigmet # Page 25

26 Assigmet #4 Page 26

27 Assigmet #5 Page 27