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Chapter 3 Real Numbers and Radicals Day 1: Roots and Radicals A2TH SWBAT evaluate radicals of any index Do Now: Simplify the following: a) 22 b) (-2)2 c) -22 d) 82 e) (-8)2 f) 53 g)(-5)3 Based on parts a and b, what are the possibilities for the β4 ? A square root of k is one of the two equal factors whose product is k. Every positive number has two square roots. Example: 1) x2 = 9 2) y2 = 5 The principal square root of a positive real number is its positive square root. In general, when referring to the square root of a number, we mean the principal square root. Example: β25 = The cube root of k is one of three equal factors whose product is k. The cube root of k is written as 3 βπ . 3 Since 4 β 4 β 4 = 64, 4 is one of the three equal factors whose product is 64 and β64 = 4. There is only one real number that is the cube of 64. This real number is called the principal cube root. Note: (β4) β (β4) β (β4) = β64. Therefore, 3 ββ64 = β4 and β 4 is the principal cube root of β64. The nth root of k is one of the n equal factors whose product is k. The nth root of k is written as k is the radicand n is the index π βπ . π βπ is the radical If k is positive, the principal nth root of k is the positive nth root If k is negative and n is odd, the principal nth root of k is negative If k is negative and n is even, there is no principal nth root of k in the set of real numbers Page 1 When a variable appears in the radicand, the radical can be simplified if the exponent of the variable is divisible by the index. Example: Since 8x6 = (2x2)(2x2)(2x2), 3 β8π₯ 6 = 2x2. Practice: 1) If a2 = 169, find all values of a . 2) Evaluate each of the following in the set of real numbers: b. ββ121π 4 a. β49 c. 3 ββ 1 d. 8 4 β81 3) Find the length of the longer leg of a right triangle if the measure of the shorter leg is 9 centimeters and the measure of the hypotenuse is 41 centimeters. 4) State whether each of the following represents a number that is rational, irrational, or neither. a. β25 b. β8 c. ββ8 d. 3 ββ8 Page 2 e. β0 f. 4 β16 g. 5 ββ243 h. β0.25 Summary The nth root of k is one of the n equal factors whose product is k. The nth root of k is written as k is the radicand n is the index π βπ . π βπ is the radical If k is positive, the principal nth root of k is the positive nth root If k is negative and n is odd, the principal nth root of k is negative If k is negative and n is even, there is no principal nth root of k in the set of real numbers Exit Ticket: For what value(s) of x is the following radical a real number? βπ β ππ Homework #1: Textbook page 87 #1, 2, 11-47 odd, 51 Page 3 Day 2: Simplifying Radicals A2TH SWBAT simplify radicals Do Now: Evaluate each of the following β 3 b) β β a) β36 8 c) β144π₯ 2 π¦10 27 d) 3 ββ125π6 π18 How would you write β12 in simplest radical form? In order to simplify a square root, we use the relationship βπ β π = βπ β βπ We usually want to use the greatest perfect square factor to write our answer in simplest radical form. Simplest radical form is achieved when the radicand is a prime number or the product of prime numbers. Example: β72 = β36 β β2 = 6β2 Simplify: a) β24π₯ 4 b) β50π3 c) β8π₯ 5 π¦ 6 d) 2β20π17 We use the same process when simplifying a cube root, breaking down the radicand using a perfect cube factor. Example: 3 3 3 3 β24 = β8 β β3 = 2 β3 Simplify: a) 3 β54 b) 3 β40π₯ 9 c) 3 β32π₯ 4 d) 3 β192π14 π12 Fractional Radicands: A radical is in simplest form when the radicand is an integer. For any non-negative a and positive c: β π π = βπ βπ Page 4 Example: β 2 3 2 3 3 3 = β × 6 = β = 9 β6 β9 = b) β 4 β6 3 Fractional Radicands continued... Simplify: a) β d) β 9π 8π3 8 9 e) 4 β 1 2 5 c) β f) 4 3 8 β 3π 2π3 π 2 More Practice! Page 5 Summary: For roots of any index: π π π βππ = βπ β βπ Exit Ticket: Simplify: a) 1 2 and π π π βπ βπ βπ = β72ππ 5 b) π 3 β375π₯ 5 π¦ 6 Homework #2: Textbook page 93 #1, 2, 3, 7, 9, 11, 17, 19, 21, 23, 27 Page 6 Day 3: Adding, Subtracting, and Multiplying Radicals A2TH SWBAT add, subtract, and multiply radicals. Do Now: Simplify: a) 1 2 β72ππ 5 b) 3 β375π₯ 5 π¦ 6 c) What are the rules when adding or subtracting algebraic terms? For example, how would you simplify (b4-5b +3) - (3b4 + b + 3)? d) What are the rules when multiply algebraic terms? For example, how would you simplify (3a2b)(2abc)? We use similar rules as above when adding, subtracting, and multiplying radical expressions. To express the sum or difference of two radicals as a single radical, the radicals must have the same index and the same radicand. In other words, they must be like radicals. 3 Example: 3 β4 3 3 + 2 β4 = 5 β4 Simplify: a) 2β5 β 4β5 b) 4β3π + 3β3π c) 3 3 β2 + 7 β2 Two radicals that do not have the same radicand or do not have the same index are unlike radicals. Example: β2 + β3 cannot be expressed as a single radical Sometimes radical expressions that seem to be unlike terms can be simplified first and then appear as like radicals. Example: β8 + β50 = 2β2 + 5β2 = 7β2 Add or Subtract: a) β27π 3 β β12π 3 1 b) β12 + 9β3 + β8 β β72 Page 7 c) 3 3 3 β8π₯ + β16π₯ + β27π₯ 1 d) 3π₯β 3π₯ + β300π₯ e) Solve for x: 4π₯ β β8 = β72 π Recall that if a and b are non-negative numbers, βππ = π π equality, βπ β βπ = π βπ β π βπ . By the symmetric property of π βππ. We can use this rule to multiply radicals. The Distributive Property may also be used when needed. Example: β8 β β2 = β16 = 4 Multiply: a) β6π3 β β18π 3 c) 5 βπ₯ β 3 4 β8π₯ 7 b) 3β2 β 4β10 4 d) β48π₯ 2 4 β β π₯2 3 e) β5 (3β6 + 12β3) f) (3 + β3 ) (2 β β6 ) g) (2 + β2 )(5 β β2 ) h) (4 + β3 ) ( 4 β β3 ) Page 8 Page 9 Page 10 Summary Page 11 Exit Ticket: Simplify: (3 β β2)2 Homework #3: Textbook page 97 #1, 3, 7, 9, 21, 25, 29, 31, 33, 39; page 100 #21-45 odd Page 12 Day 4: Dividing Radicals and Rationalizing a Denominator A2TH SWBAT 1) simplify quotients of radical expressions and 2) simplify radical expressions by rationalizing the denominator. Do Now: Simplify the following radical expressions β a) 3 3 3 3 β2 β56 + 5 β189 β 3 β7 b) ( β2 3 β 7)( β4 β 7) Compare these two computations: β9 β25 β The general rule for the quotient of roots: π βπ π βπ π π Simplify: a) 3 β 5 27 β4 β9 = π If βπ and βπ are real numbers, with βπ β 0, then π π = β π This rule can also be used in the reversed order: Example: 9 25 2 π π βπ = Example: 3 b) β 8π₯ 2 49 β8 β2 π βπ π βπ = β 8 2 = β4 = 2 c) β75π₯ 3 π¦ β3π₯ Page 13 A fraction is in simplest form when the denominator is a rational number. We do not want a radical in the denominator. 2 Example 1. Rationalize: β7 2 β7 β7 β3π₯ Simplify: a) β2 b) Example 2. Rationalize: ( 7) = β 2β7 7 3 5β10 c) β5π₯ 3 4 ββ256 β80 d) 4 β5π₯ 4 3 ββ2 3 2+ β6 Here we use whatβs called the conjugate to rationalize the denominator. The conjugate of (2 + β6) is (2 β β6). 3 2+ β6 Rationalize: a) b) c) ( 2β β6 6β3β6 2β β6 4β2β6+2β6β β36 )= = 6β3β6 4β6 = 6β3β6 β2 = β 6β3β6 2 5 β11β7 6 β5+ β2 β3+3 5β β3 Page 14 d) βπ₯+ βπ¦ βπ₯β βπ¦ e) βπ₯+π¦ 3+ βπ₯+π¦ f) g) 6 β3 β5 β3π₯ + β h) Solve for x: 8 β2 β3 β5π₯ π₯β2 = 3 β π₯ Page 15 Page 16 Summary Exit Ticket: Homework #4 : Textbook page 103 #17-31 odd; page 107 #21-35 odd, 41-45 odd Page 17 Day 5: Solving Radical Equations A2TH SWBAT: solve radical equations Do Now: A radical equation is an equation in which the variable is hiding inside a radical sign. To solve a radical equation, follow these steps: 1. Isolate the radical (or one of the radicals) to one side of the equal sign. 2. If the radical is a square root, square each side of the equation. (If the radical is not a square root, raise each side to a power equal to the index of the root.) 3. Solve the resulting equation. 4. Check your answer(s) to avoid extraneous roots. 1. xο2 ο7 ο½ 2 2. β2π₯ β1 +5 = 2 Page 18 3. 4 + β1 β 3π₯ = 12 5. β5π₯ 7. π₯ + 3 = β3π₯ + 7 β 3 = β30 β 2π₯ 4. 6. π₯ 20 ο 2 x ο½ 5x ο 8 β 1 = β5π₯ β 9 8. π₯ = 1 + β15 β 7π₯ Page 19 9. 2βπ₯ 11. + 8 = 3βπ₯ β 2 2 ο« 3x ο 2 ο½ 6 3 13. βπ₯ β 4 = β9π₯ 10. βπ₯ 12. + 5 = βπ₯ 2 β 15 5 β3π₯ β 5 = β2 14. βπ₯ β βπ₯ β 5 = 1 Page 20 15.β2π₯ + 3 = 5 β βπ₯ + 1 16. β2π₯ + 1 β βπ₯ β 3 = 2 Summary To solve radical equations: 1. Isolate the radical (or one of the radicals) to one side of the equal sign. 2. If the radical is a square root, square each side of the equation. (If the radical is not a square root, raise each side to a power equal to the index of the root.) 3. Solve the resulting equation. 4. Check your answer(s) to avoid extraneous roots. Exit Ticket: Homework #5: page 112 #2, 11-35 odd Page 21 A2TH Packet #3: Name:______________________________ Teacher:____________________________ Pd: _______ Page 22 Table of Contents o Day 1: SWBAT: evaluate radicals of any index HW: Textbook page 87 #1, 2, 11-47 odd, 51 o Day 2: SWBAT: simplify radicals HW: Textbook page 93 #1, 2, 3, 7, 9, 11, 17, 19, 21, 23, 27 o Day 3: SWBAT: add, subtract and multiply radicals HW: Textbook page 97 #1,3,7,9,21,25,29,31,33,39; page 100 #21 β 45 odd o Day 4: SWBAT: 1) simplify quotients of radical expressions and 2) simplify radical expressions by rationalizing the denominator. HW: Textbook page 103 #17-31 odd; page 107 #21-35 odd, 41-45 odd o Day 5: solve radical equations HW: page 112 #2, 11-35 odd Page 23 Assignment #1 Assignment #2 Page 24 Assignment #3 Page 25 Assignment #4 Page 26 Assignment #5 Page 27