7.3 Simplifying Radical Expressions

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1 6 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers 11. = /5 -/5 1. y = -/ y/, or y Use a calculator to write a four-decimal-place approimation of each number / / / /7 17. In physics, the speed of a wave traveling over a stretched string with tension t and density u is given by the epression t. Write this epression with rational eponents. u 18. In electronics, the angular frequency of oscillations in a certain type of circuit is given by the epression 1LC -1/. Use radical notation to write this epression. 7. Simplifying Radical Epressions S 1 Use the Product Rule for Radicals. Use the Quotient Rule for Radicals. Simplify Radicals. Use the Distance and Midpoint Formulas. 1 Using the Product Rule It is possible to simplify some radicals that do not evaluate to rational numbers. To do so, we use a product rule and a quotient rule for radicals. To discover the product rule, notice the following pattern. 9 # = # = 6 9 # = 6 = 6 Since both epressions simplify to 6, it is true that 9 # = 9 # This pattern suggests the following product rule for radicals. Product Rule for Radicals If n a and n b are real numbers, then n a # n b = n ab Notice that the product rule is the relationship a 1/n # b 1/n = 1ab 1/n stated in radical notation. EXAMPLE 1 Multiply. a. # 5 b. 1 # c. # d. 5y # e. # b A a A a. # 5 = # 5 = 15 b. 1 # = 1 c. # = # = 8 = d. 5y # = 5y # = 10y e. # b A a A = # b A a = b A a 1 Multiply. a. 5 # 7 b. 1 # z c. 15 # 5 d. 5y # 5 e. # t A m A

2 Section 7. Simplifying Radical Epressions 7 Using the Quotient Rule To discover a quotient rule for radicals, notice the following pattern. A 9 = 9 = Since both epressions simplify to, it is true that A 9 = 9 This pattern suggests the following quotient rule for radicals. Quotient Rule for Radicals If n a and n b are real numbers and n b is not zero, then n a A b = n a n b Notice that the quotient rule is the relationship a a 1/n b b = a1/n stated in radical 1/n b notation. We can use the quotient rule to simplify radical epressions by reading the rule from left to right or to divide radicals by reading the rule from right to left. For eample, A 16 = 16 = 75 = A 75 = 5 = 5 Using A n a b = n a n b Using n a n b = n a A b Note: Recall that from Section 7. on, we assume that variables represent positive real numbers. Since this is so, we need not insert absolute value bars when we simplify even roots. EXAMPLE 5 a. A 9 a. A 5 9 = 5 9 = 5 7 c. A 8 7 = 8 7 = Use the quotient rule to simplify. b. A 9 c. A 8 7 b. A 9 = 9 = d. B 16y d. A 16y = 16y = y Use the quotient rule to simplify. a. A 6 9 b. A z 16 c. A 15 8 d. A

3 8 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers Simplifying Radicals Both the product and quotient rules can be used to simplify a radical. If the product rule is read from right to left, we have that n ab = n a # n b. This is used to simplify the following radicals. EXAMPLE Simplify the following. a. 50 b. c. 6 d. a. Factor 50 such that one factor is the largest perfect square that divides 50. The largest perfect square factor of 50 is 5, so we write 50 as 5 # and use the product rule for radicals to simplify. b. = 8 # = 8 # = 50 = 5 # = 5 # = 5 The largest perfect square factor of 50 c The largest perfect cube factor of c. 6 The largest perfect square factor of 6 is 1, so 6 cannot be simplified further. d. = 16 # = 16 # = c c The largest fourth power factor of Simplify the following. a. 98 b. 5 c. 5 d. Helpful Hint Don t forget that, for eample, 5 means 5 #. After simplifying a radical such as a square root, always check the radicand to see that it contains no other perfect square factors. It may, if the largest perfect square factor of the radicand was not originally recognized. For eample, 00 = # 50 = # 50 = 50 Notice that the radicand 50 still contains the perfect square factor 5. This is because is not the largest perfect square factor of 00. We continue as follows. 50 = 5 # = # 5 # = # 5 # = 10 The radical is now simplified since contains no perfect square factors (other than 1)., 9, 16, 5, 6, 9, 6, 81, 100, 11, 1 8, 7, 6, 15 Helpful Hint To help you recognize largest perfect power factors of a radicand, it will help if you are familiar with some perfect powers. A few are listed below. Perfect Squares 1, Perfect Cubes 1, 1 5 Perfect Fourth 1, 16, 81, 56 Powers 1 In general, we say that a radicand of the form n a is simplified when the radicand a contains no factors that are perfect nth powers (other than 1 or -1).

4 Section 7. Simplifying Radical Epressions 9 EXAMPLE Use the product rule to simplify. a. 5 b. 5 6 y 8 c. 81z 11 a. 5 = 5 # = 5 # = 5 b. 5 6 y 8 = 7 # # 6 # y 6 # y = 7 6 y 6 # y = 7 6 y 6 # y = y y c. 81z 11 = 81 # z 8 # z = 81z 8 # z = z z Find the largest perfect square factor. Apply the product rule. Factor the radicand and identify perfect cube factors. Apply the product rule. Factor the radicand and identify perfect fourth power factors. Apply the product rule. Use the product rule to simplify. a. 6z 7 b. p q 7 c EXAMPLE 5 a. 0 5 a. 0 5 = 0 A 5 = = b. 50 Use the quotient rule to divide, and simplify if possible. c. 7 8 y 8 d. a 8 b 6 6y a -1 b Apply the quotient rule. b. 50 = 1 # 50 A = 1 # 5 = 1 # 5 # = 1 # 5 # Apply the quotient rule. Factor 5. = 5 c. 7 8 y 8 6y = 7 # B 8 y8 6y = 7 # 8 y 6 = 7 8 y 6 # = 7 # 8 y 6 # (Continued on net page) Apply the quotient rule. Factor. Apply the product rule.

5 0 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers = 7 # y # = 1y d. a 8 b 6 a 8 b 6 = a -1 b B a -1 b = a 9 b = 16 # a 8 # b # # a = 16a 8 b # a = # a b # a = a b a 5 Use the quotient rule to divide and simplify. a b. 98z c y 7 5y d y y CONCEPT CHECK Find and correct the error: 7 9 = 7 A 9 = Using the Distance and Midpoint Formulas ( 1, y 1 ) y (, y ) d b y y 1 Now that we know how to simplify radicals, we can derive and use the distance formula. The midpoint formula is often confused with the distance formula, so to clarify both, we will also review the midpoint formula. The Cartesian coordinate system helps us visualize a distance between points. To find the distance between two points, we use the distance formula, which is derived from the Pythagorean theorem. To find the distance d between two points 1 1, y 1 and 1, y as shown to the left, notice that the length of leg a is - 1 and that the length of leg b is y - y 1. Thus, the Pythagorean theorem tells us that a 1 d = a + b or d = y - y 1 or d = y - y 1 This formula gives us the distance between any two points on the real plane. Distance Formula The distance d between two points 1 1, y 1 and 1, y is given by d = y - y 1 Answer to Concept Check: 7 9 = = 1 EXAMPLE 6 Find the distance between 1, -5 and 11, -. Give an eact distance and a three-decimal-place approimation. To use the distance formula, it makes no difference which point we call 1 1, y 1 and which point we call 1, y. We will let 1 1, y 1 = 1, -5 and 1, y = 11, -.

6 Section 7. Simplifying Radical Epressions 1 d = y - y 1 = [ ] = = = 1.1 The distance between the two points is eactly units, or approimately 1.1 units. 6 Find the distance between 1 -, 7 and 1 -,. Give an eact distance and a three-decimal-place approimation. P y Q M The midpoint of a line segment is the point located eactly halfway between the two endpoints of the line segment. On the graph to the left, the point M is the midpoint of line segment PQ. Thus, the distance between M and P equals the distance between M and Q. Note: We usually need no knowledge of roots to calculate the midpoint of a line segment. We review midpoint here only because it is often confused with the distance between two points. The -coordinate of M is at half the distance between the -coordinates of P and Q, and the y-coordinate of M is at half the distance between the y-coordinates of P and Q. That is, the -coordinate of M is the average of the -coordinates of P and Q; the y-coordinate of M is the average of the y-coordinates of P and Q. Midpoint Formula The midpoint of the line segment whose endpoints are 1 1, y 1 and 1, y is the point with coordinates a 1 +, y 1 + y b EXAMPLE 7 Find the midpoint of the line segment that joins points P1 -, and Q(1, 0). Use the midpoint formula. It makes no difference which point we call 1 1, y 1 or which point we call 1, y. Let 1 1, y 1 = 1 -, and 1, y = 11, 0. midpoint = a 1 + = a = a -, b = a -1, b, y 1 + y b, + 0 b P(, ) ( 1, w) y Q(1, 0) The midpoint of the segment is a -1, b. 7 Find the midpoint of the line segment that joins points P15, - and Q18, -6.

distance measured in units midpoint it is a point Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Some choices may be used more than once. distance midpoint point 1.

7 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers Helpful Hint The distance between two points is a distance. The midpoint of a line segment is the point halfway between the endpoints of the segment. distance measured in units midpoint it is a point Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Some choices may be used more than once. distance midpoint point 1. The of a line segment is a eactly halfway between the two endpoints of the line segment.. The between two points is a distance, measured in units.. The formula is d = y - y 1.. The formula is a 1 +, y 1 + y b. Martin-Gay Interactive Videos See Video 7. Watch the section lecture video and answer the following questions From Eample 1 and the lecture before, in order to apply the product rule for radicals, what must be true about the indees of the radicals being multiplied? 6. From Eamples 6, when might you apply the quotient rule (in either direction) in order to simplify a fractional radical epression? 7. From Eample 8, we know that an even power of a variable is a perfect square factor of the variable, leaving no factor in the radicand once simplified. Therefore, what must be true about the power of any variable left in the radicand of a simplified square root? Eplain. 8. From Eample 10, the formula uses the coordinates of two points similar to the slope formula. What caution should you take when replacing values in the formula? 9. Based on Eample 11, complete the following statement. The -value of the midpoint is the of the -values of the endpoints and the y-value of the midpoint is the of the y-values of the endpoints. 7. Eercise Set Use the product rule to multiply. See Eample #. 11 # #. 7 # 5. # # 5 7. # 8. y # # # n A A y A m A # 5 1. ab # 7ab Use the quotient rule to simplify. See Eamples and A 9 A 9 B16 A A 81 5 A 11 y A81 A6

8 Section 7. Simplifying Radical Epressions a A 8. B 81. A81y 1 y B 100 A B y z 6 5 y 10 B y 8. B B z 7 6a A b 9 See Eamples and y y 7. 6y 9. a 8 b 7. 5 z 1 5. y 5 6. y a b y y z y a 8 b ab y r 9 s r 9 s a 6 b y y 0 Use the quotient rule to divide. Then simplify if possible. See Eample y y 8 5m 7 m a 10 b a b a 7 b 6 a b 18-70y 5y y 5 y y 5 y y y - Find the distance between each pair of points. Give an eact distance and a three-decimal-place approimation. See Eample (5, 1) and (8, 5) 78. (, ) and (1, 8) , and 11, , - and 1 -, , and 1-8, , - and 1-6, , - 1 and 11, , 0 and 10, , -.6 and 1-8.6, (9.6,.5) and 1-1.9, -.7 Find the midpoint of the line segment whose endpoints are given. See Eample , -8, 1, 88. (, 9), (7, 11) , -1, 1-8, , -, 16, ,, 1-1, , 5, 1-1, 6 9. a 1, 8 b, a -, 5 8 b 9. a - 5, 7 15 b, a - 5, - 15 b , 15, 11, , - 11, 11, , -.5, 17.8, ,.1, 1-6.7, 1.9 REVIEW AND PREVIEW Perform each indicated operation. See Sections 1. and (6)(8) y - 8y y 1-8y CONCEPT EXTENSIONS Answer true or false. Assume all radicals represent nonzero real numbers n a # n b = n ab, # 11 = 18, # 11 = 77, y 8 = 7 # y 8, 11. n a n b = A n a b, = 8,

CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers Find and correct the error. See the Concept Check in this section. 115. 116.

9 CHAPTER 7 Rational Eponents, Radicals, and Comple Numbers Find and correct the error. See the Concept Check in this section = 6 A6 = 1 = 1 16 = 16 A = See a Concept Check in this section. Assume variables represent positive numbers y a 1 b c a 9 b 1 c 11. z q 17 r 0 s 7 1. p 11 q r The formula for the radius r of a sphere with surface area A A is given by r =. Calculate the radius of a standard A p zorb whose outside surface area is.17 sq m. Round to the nearest tenth. (A zorb is a large inflated ball within a ball in which a person, strapped inside, may choose to roll down a hill. Source: Zorb, Ltd.) a. Approimate to one decimal place the demand per week of an older released DVD if the rental price is $ per two-day rental. b. Approimate to one decimal place the demand per week of an older released DVD if the rental price is $5 per two-day rental. c. Eplain how the owner of the store can use this equation to predict the number of copies of each DVD that should be in stock. 17. The formula for the lateral surface area A of a cone with height h and radius r is given by A = prr + h a. Find the lateral surface area of a cone whose height is centimeters and whose radius is centimeters. b. Approimate to two decimal places the lateral surface area of a cone whose height is 7. feet and whose radius is 6.8 feet. h r 18. Before Mount Vesuvius, a volcano in Italy, erupted violently in 79 c.e., its height was 190 feet. Vesuvius was roughly cone-shaped, and its base had a radius of approimately 5,00 feet. Use the formula for the lateral surface area of a cone, given in Eercise 17, to approimate the surface area this volcano had before it erupted. (Source: Global Volcanism Network) 16. The owner of Knightime Classic Movie Rentals has determined that the demand equation for renting older released DVDs is F1 = , where is the price in dollars per two-day rental and F1 is the number of times the DVD is demanded per week. 190 ft 5,00 ft 7. Adding, Subtracting, and Multiplying Radical Epressions S 1 Add or Subtract Radical Epressions. Multiply Radical Epressions. 1 Adding or Subtracting Radical Epressions We have learned that sums or differences of like terms can be simplified. To simplify these sums or differences, we use the distributive property. For eample, + = 1 + = 5 and 7 y - y = 17 - y = y The distributive property can also be used to add like radicals. Like Radicals Radicals with the same inde and the same radicand are like radicals.

6.2 Adding and Subtracting Rational Expressions

6.2 Adding and Subtracting Rational Expressions 8 CHAPTER 6 Rational Epressions Simplify. Assume that no denominator is 0. 99. p - - p 00. + q n q n + n + k - 9 0. n 0. - 6 + k Perform the indicated operation. Write all answers in lowest terms. 0. 0.