10-3 Operations with Radical Expressions. Simplify each expression. 1. SOLUTION: 2. SOLUTION: 3. SOLUTION: 4. SOLUTION: 5. SOLUTION: 6.

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1 Simplify each expression Page 1

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3 GEOMETRY The area A of a triangle can be found by using the formula, where b represents the base and h is the height. What is the area of the triangle shown? Page 3

4 13. GEOMETRY The area A of a triangle can be found by using the formula, where b represents the base and h is the height. What is the area of the triangle shown? The area of the triangle is. Simplify each expression Page 4

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7 GEOMETRY Find the perimeter and area of a rectangle with a width of and a length of. Page 7

8 6. GEOMETRY Find the perimeter and area of a rectangle with a width of and a length of. The perimeter is units. The area is 1 square units. Simplify each expression Page 8

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11 ROLLER COASTERS The velocity v in feet per second of a roller coaster at the bottom of a hill is related to the vertical drop h in feet and the velocity v of the coaster at the top of the hill by the formula. a. What velocity must a coaster have at the top of a 5-foot hill to achieve a velocity of 1 feet per second at the bottom? b. Explain why v = v is not equivalent to the formula given. a. Let v = 1 and h = 5. The coaster must have a velocity of feet per second at the top of the hill. b. Sample answer: In the formula given, we are taking the square root of the difference, not the square root of each term. 34. FINANCIAL LITERACY Tadi invests $5 in a savings account. In two years, Tadi has $3 in his account. You can use the formula to find the average annual interest rate r that the account has earned. The initial investment is v and v is the amount in two years. What was the average annual interest rate that Tadi s account earned? Let v = 5 and let v = 3. Page 11

12 The coaster must have a velocity of feet per second at the top of the hill.. Sample answer: the formula given, we are taking the square root of the difference, not the square root of each 1-3bOperations with In Radical Expressions term. 34. FINANCIAL LITERACY Tadi invests $5 in a savings account. In two years, Tadi has $3 in his account. You can use the formula to find the average annual interest rate r that the account has earned. The initial investment is v and v is the amount in two years. What was the average annual interest rate that Tadi s account earned? Let v = 5 and let v = 3. The average annual interest rate that Tadi s account earned was about 1.5%. 35. ELECTRICITY Electricians can calculate the electrical current in amps A by using the formula, where w is the power in watts and r the resistance in ohms. How much electrical current is running through a microwave oven that has 85 watts of power and 5 ohms of resistance? Write the number of amps in simplest radical form, and then estimate the amount of current to the nearest tenth. Let w = 85 and let r = 5. There are or about 13 amps of electrical current running through the microwave oven. 36. CHALLENGE Determine whether the following statement is true or false. Provide a proof or counterexample to support your answer. x +y > when x > and y > Because x and y are both positive, each side of the inequality represents a positive number. So, you can prove the statement by squaring each side of the inequality. Page 1

13 There are or about 13 amps of electrical current running through the microwave oven. 36. CHALLENGE Determine whether the following statement is true or false. Provide a proof or counterexample to support your answer. x +y > when x > and y > Because x and y are both positive, each side of the inequality represents a positive number. So, you can prove the statement by squaring each side of the inequality. Because x > and y >, the product xy must be a positive number.thus, xy > is always true. Therefore, is a true statement for all x > and y >. 37. CCSS ARGUMENTS Make a conjecture about the sum of a rational number and an irrational number. Is the sum rational or irrational? Is the product of a nonzero rational number and an irrational number rational or irrational? Explain your reasoning. Examine the sum of several pairs of rational and irrational numbers: is in lowest terms, and is irrational. is in lowest terms and is irrational. is irrational. Examine the product of several pairs of non-zero rational and irrational numbers: is irrational is irrational is irrational. From the above examples, we should come up with the conjecture that the sum of a rational number and an irrational number is irrational, and the product of a rational number and an irrational number is irrational. Page OPEN ENDED Write an equation that shows a sum of two radicals with different radicands. Explain how you

14 Therefore, is a true statement for all x > and y >. 37. CCSS ARGUMENTS Make a conjecture about the sum of a rational number and an irrational number. Is the sum rational or irrational? Is the product of a nonzero rational number and an irrational number rational or irrational? Explain your reasoning. Examine the sum of several pairs of rational and irrational numbers: is in lowest terms, and is irrational. is in lowest terms and is irrational. is irrational. Examine the product of several pairs of non-zero rational and irrational numbers: is irrational is irrational is irrational. From the above examples, we should come up with the conjecture that the sum of a rational number and an irrational number is irrational, and the product of a rational number and an irrational number is irrational. 38. OPEN ENDED Write an equation that shows a sum of two radicals with different radicands. Explain how you could combine these terms. Sample answer: When you simplify, you get same radicand, you can add them.. When you simplify, you get. Because these two numbers have the 39. WRITING IN MATH Describe step by step how to multiply two radical expressions, each with two terms. Write an example to demonstrate your description. For example, you can use the FOIL method. You multiply the first terms within the parentheses. Then you multiply the outer terms within the parentheses. Then you would multiply the inner terms within the parentheses. And, then you would multiply the last terms within each parentheses. Combine any like terms and simplify any radicals. For example: Page 14

15 you simplify, youexpressions get. When you simplify 1-3When Operations with Radical same radicand, you can add them., you get. Because these two numbers have the 39. WRITING IN MATH Describe step by step how to multiply two radical expressions, each with two terms. Write an example to demonstrate your description. For example, you can use the FOIL method. You multiply the first terms within the parentheses. Then you multiply the outer terms within the parentheses. Then you would multiply the inner terms within the parentheses. And, then you would multiply the last terms within each parentheses. Combine any like terms and simplify any radicals. For example: 4. SHORT RESPONSE The population of a town is 13, and is increasing by about 5 people per year. This can be represented by the equation p = 13, + 5y, where y is the number of years from now and p represents the population. In how many years will the population of the town be 14,5? Let p = 14,5. In 6 years, the population of the town will be 14, GEOMETRY Which expression represents the sum of the lengths of the 1 edges on this rectangular solid? A (a + b + c) B 3(a + b + c) C 4(a + b + c) D 1(a + b + c) There are 4 edges that have a length of a, 4 edges that have a length of b, and 4 edges that have a length of c. So, the expression 4a + 4b + 4c or 4(a + b + c) represents the sum of the lengths of the 1 edges of the rectangular solid. Choice C is the correct answer. 4. Evaluate and F 4 4 G 4 for n = 5. Page 15

16 There are 4 edges that have a length of a, 4 edges that have a length of b, and 4 edges that have a length of c. So, the expression 4a + 4b + 4c or 4(a + b + c) represents the sum of the lengths of the 1 edges of the rectangular solid. Choice C is the correct answer. 4. Evaluate and for n = 5. F 4 4 G 4 H ; 4 J ; Substitute 5 for n in both expressions. Choice G is correct. 43. The current I in a simple electrical circuit is given by the formula, where V is the voltage and R is the resistance of the circuit. If the voltage remains unchanged, what effect will doubling the resistance of the circuit have on the current? A The current will remain the same. B The current will double its previous value. C The current will be half its previous value. D The current will be two units more than its previous value. The current and the resistance have an inverse relationship. If the resistance is double and the voltage remains the same, the current will be half of its previous value. Consider an example with the resistance is 4 and voltage is 1. Find I when R doubles. Choice C is the correct answer. Simplify. 44. Page 16

17 Choice C is the correct answer. Simplify Page 17

18 Graph each function. Compare to the parent graph. State the domain and range. 5. x y is multiplied by a value greater than 1, so the graph is a vertical stretch of. The parent function Another way to identify the stretch is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x x } and the range is {y y }. 51. x y Page 18

19 is multiplied by a value greater than 1, so the graph is a vertical stretch of. The parent function Another way to identify the stretch is to notice that the y-values in the table are times the corresponding y-values 1-3for Operations Radical Expressions the parent with function. The domain is {x x } and the range is {y y }. 51. x y The parent function is multiplied by a value less than 1, so the graph is a vertical stretch of and a reflection across the x-axis. Another way to identify the stretch is to notice that the y-values in the table are 3 times the corresponding y-values for the parent function. The domain is {x x } and the range is {y y }. 5. x 1 y The value 1 is being added to the square root of the parent function, so the graph is translated 1 unit left from the parent graph. Another way to identify the translation is to note that the x-values in the table are 1 less than the corresponding x-values for the parent function. The domain is {x x 1}, and the range is {y y }. 53. x 4 y Page 19

20 The value 1 is being added to the square root of the parent function, so the graph is translated 1 unit left the parent graph. Another way to identify the translation is to note that the x-values in the table are 1 1-3from Operations with Radical Expressions less than the corresponding x-values for the parent function. The domain is {x x 1}, and the range is {y y }. 53. x 4 y The value 4 is being subtracted from the square root of the parent function, so the graph is translated 4 units right from the parent graph. Another way to identify the translation is to note that the x-values in the table are 4 more than the corresponding x-values for the parent function. The domain is {x x 4}, and the range is {y y }. 54. x y The value 3 is being added to the parent function 4 5, so the graph is translated up 3 units from the parent graph. Another way to identify the translation is to note that the y-values in the table are 3 greater than the corresponding y-values for the parent function. The domain is {x x } and the range is {y y 3}. 55. x y Page

21 The value 3 is being added to the parent function, so the graph is translated up 3 units from the parent graph. Another to identify the translation is to note that the y-values in the table are 3 greater than the 1-3 Operations withway Radical Expressions corresponding y-values for the parent function. The domain is {x x } and the range is {y y 3}. 55. x y The value is being subtracted from the parent function 4, so the graph is translated down units from the parent graph. Another way to identify the translation is to note that the y-values in the table are less than the corresponding y-values for the parent function. The domain is {x x } and the range is {y y }. Factor each trinomial. 56. x + 1x y + 13y p 17p x + 6x 7 6. y y 4 Page 1

22 59. x + 6x 7 6. y y w + w 6. FINANCIAL LITERACY Determine the value of an investment if $4 is invested at an interest rate of 7.5% compounded quarterly for 7 years. Use the formula for calculating compound interest. The value of the investment after 7 years is about $ Solve each equation. Round each solution to the nearest tenth, if necessary c 1. = q 33.7 = m + 4 = 9.6 esolutions Manual - Powered by Cognero 66. Page

23 t + 6.5t = 8 Page 3