10-3 Operations with Radical Expressions Simplify each expression. 1.  SOLUTION:  

2.  SOLUTION:  

3.  SOLUTION:  

4.  SOLUTION:  

5.  SOLUTION:  

6.  SOLUTION:  

eSolutions Manual - Powered by Cognero

Page 1

10-3 Operations with Radical Expressions

6.  SOLUTION:  

7.  SOLUTION:  

8.  SOLUTION:  

9.  SOLUTION:   eSolutions Manual - Powered by Cognero

Page 2

10-3 Operations with Radical Expressions

9.  SOLUTION:  

10.  SOLUTION:  

11.  SOLUTION:  

12.  SOLUTION:  

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?

  eSolutions Manual - Powered by Cognero

Page 3

10-3 Operations with Radical Expressions

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?

 

SOLUTION:  

  The area of the triangle is

.

Simplify each expression. 14.  SOLUTION:  

15.  SOLUTION:  

16.  SOLUTION:   eSolutions Manual - Powered by Cognero

Page 4

SOLUTION:   10-3 Operations with Radical Expressions

16.  SOLUTION:  

17.  SOLUTION:  

18.  SOLUTION:  

19.  SOLUTION:  

20.  eSolutions Manual - Powered by Cognero SOLUTION:  

Page 5

10-3 Operations with Radical Expressions

20.  SOLUTION:  

21.  SOLUTION:  

22.  SOLUTION:  

23.  SOLUTION:  

eSolutions Manual - Powered by Cognero

Page 6

10-3 Operations with Radical Expressions

23.  SOLUTION:  

24.  SOLUTION:  

25.  SOLUTION:  

26. GEOMETRY  Find the perimeter and area of a rectangle with a width of SOLUTION:  

eSolutions Manual - Powered by Cognero

 and a length of 

. Page 7

10-3 Operations with Radical Expressions

26. GEOMETRY  Find the perimeter and area of a rectangle with a width of

 and a length of 

.

SOLUTION:  

The perimeter is

 units.

The area is 12 square units. Simplify each expression. 27.  SOLUTION:  

eSolutions Manual - Powered by Cognero

28. 

Page 8

10-3 Operations with Radical Expressions

28.  SOLUTION:  

29.  SOLUTION:  

30.  SOLUTION:  

eSolutions Manual - Powered by Cognero

Page 9

10-3 Operations with Radical Expressions

30.  SOLUTION:  

31.  SOLUTION:  

32.  SOLUTION:  

eSolutions Manual - Powered by Cognero

Page 10

10-3 Operations with Radical Expressions

32.  SOLUTION:  

33. 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 v0 of the coaster at the top of the hill by the formula

.

  a.  What velocity must a coaster have at the top of a 225-foot hill to achieve a velocity of 120 feet per second at the bottom?

  b.  Explain why v0 = v −

 is not equivalent to the formula given.

SOLUTION:   a. Let v = 120 and h = 225.

 

  The coaster must have a velocity of 0 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 $225 in a savings account. In two years, Tadi has $232 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 v0 and v2 is the amount in two years. What was the average annual interest rate that Tadi’s account earned? SOLUTION:   Let v0 = 225 and let v2 = 232.

 

eSolutions Manual - Powered by Cognero

Page 11

  The coaster must have a velocity of 0 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 10-3bOperations with In Radical Expressions term. 34. FINANCIAL LITERACY  Tadi invests $225 in a savings account. In two years, Tadi has $232 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 v0 and v2 is the amount in two years. What was the average annual interest rate that Tadi’s account earned? SOLUTION:   Let v0 = 225 and let v2 = 232.

 

  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 850 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. SOLUTION:   Let w = 850 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 > 0 and y > 0

SOLUTION:  

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.   eSolutions Manual - Powered by Cognero Page 12

10-3  Operations with Radical Expressions 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 > 0 and y > 0

SOLUTION:  

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 > 0 and y > 0, the product 2xy must be a positive number.Thus, 2xy > 0 is always true.   Therefore,    

is a true statement for all x > 0 and y > 0.

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. SOLUTION:   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.    eSolutions Manual - Powered by Cognero Page 13   38. OPEN ENDED  Write an equation that shows a sum of two radicals with different radicands. Explain how you

  Therefore, is a true statement for all x > 0 and y > 0.   10-3 Operations with Radical Expressions   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. SOLUTION:   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. SOLUTION:   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. SOLUTION:   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:   eSolutions Manual - Powered by Cognero Page 14  

you simplify , youExpressions get . When you simplify 10-3When 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. SOLUTION:   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:     40. SHORT RESPONSE  The population of a town is 13,000 and is increasing by about 250 people per year. This can be represented by the equation p = 13,000 + 250y, 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 14,500? SOLUTION:   Let p = 14,500.

 

  In 6 years, the population of the town will be 14,500. 41. GEOMETRY  Which expression represents the sum of the lengths of the 12 edges on this rectangular solid?

 

  A  2(a + b + c)

  B  3(a + b + c)

  C  4(a + b + c)

  D  12(a + b + c) SOLUTION:   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 12 edges of the rectangular solid.    Choice C is the correct answer. and 42. Evaluate   eSolutions Manual - Powered by Cognero F  4; 4   G  4; 2

for n = 25. Page 15

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 12 edges of the rectangular solid.  10-3  Operations with Radical Expressions Choice C is the correct answer. 42. Evaluate   F  4; 4   G  4; 2   H 2; 4   J 2; 2

and

for n = 25.

SOLUTION:   Substitute 25 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. SOLUTION:   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 100. Find I when R doubles. 

 

  Choice C is the correct answer. eSolutions Manual - Powered by Cognero

Simplify.

44. 

Page 16

10-3  Operations with Radical Expressions Choice C is the correct answer. Simplify. 44.  SOLUTION:  

45.  SOLUTION:  

46.  SOLUTION:  

47.  SOLUTION:  

48.  SOLUTION:   eSolutions Manual - Powered by Cognero

Page 17

10-3 Operations with Radical Expressions

48.  SOLUTION:  

49.  SOLUTION:  

Graph each function. Compare to the parent graph. State the domain and range. 50.  SOLUTION:   x 0 y 0

0.5 ≈ 1.4

1 2

2 ≈ 2.8

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 2 times the corresponding y-values for the parent function.  The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 51.  SOLUTION:   x 0 y 0

0.5 ≈ –2.1

eSolutions Manual - Powered by Cognero

1 –3

2 ≈ –4.2

Page 18

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 2 times the corresponding y-values 10-3for the parent function.  The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. Operations with Radical Expressions 51.  SOLUTION:   x 0 y 0

0.5 ≈ –2.1

1 –3

2 ≈ –4.2

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 ≥ 0} and the range is {y|y ≤ 0}. 52.  SOLUTION:   x –1 y 0

–0.5 ≈ 0.7

0 1

1 ≈ 1.4

2 ≈ 1.7

3 ≈ 2

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 ≥ 0}. 53.  SOLUTION:   x 4 y 0

4.5 ≈ 0.7

eSolutions Manual - Powered by Cognero

5 1

6 ≈ 1.4

7 ≈ 1.7

8 2

Page 19

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 10-3from 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 ≥ 0}. 53.  SOLUTION:   x 4 y 0

4.5 ≈ 0.7

5 1

6 ≈ 1.4

7 ≈ 1.7

8 2

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 ≥  0}. 54.  SOLUTION:   x 0 y 3

0.5 ≈ 3.7

1 4

2 ≈ 4.4

3 ≈ 4.7

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 ≥ 0} and the range is {y|y ≥ 3}. 55.  SOLUTION:   x 0 y –2

0.5 ≈ –1.3

eSolutions Manual - Powered by Cognero

1 –1

2 ≈ –0.6

3 ≈ –0.3

4 0

Page 20

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 10-3 Operations withway Radical Expressions corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 3}. 55.  SOLUTION:   x 0 y –2

0.5 ≈ –1.3

1 –1

2 ≈ –0.6

3 ≈ –0.3

The value 2 is being subtracted from the parent function

4 0

, so the graph is translated down 2 units from the

parent graph . Another way to identify the translation is to note that the y-values in the table are 2 less than the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}. Factor each trinomial. 2

56. x + 12x + 27 SOLUTION:  

2

57. y + 13y + 30 SOLUTION:  

2

58. p − 17p + 72 SOLUTION:  

2

59. x + 6x – 7 SOLUTION:  

2

60. y − y − 42 SOLUTION:   eSolutions Manual - Powered by Cognero

2

Page 21

59. x + 6x – 7 SOLUTION:   10-3 Operations with Radical Expressions 2

60. y − y − 42 SOLUTION:  

2

61. −72 + 6w + w

SOLUTION:  

62. FINANCIAL LITERACY  Determine the value of an investment if $400 is invested at an interest rate of 7.25% compounded quarterly for 7 years. SOLUTION:   Use the formula for calculating compound interest.

The value of the investment after 7 years is about $661.44. Solve each equation. Round each solution to the nearest tenth, if necessary. 63. −4c − 1.2 = 0.8 SOLUTION:  

64. −2.6q − 33.7 = 84.1 SOLUTION:  

65. 0.3m + 4 = 9.6 SOLUTION:  

eSolutions Manual - Powered by Cognero 66. 

SOLUTION:  

Page 22

SOLUTION:   10-3 Operations with Radical Expressions

66.  SOLUTION:  

67.  SOLUTION:  

68. 3.6t + 6 − 2.5t = 8 SOLUTION:  

eSolutions Manual - Powered by Cognero

Page 23