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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
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10-3 Operations with Radical Expressions
6. SOLUTION: 7. SOLUTION: 8. SOLUTION: 9. SOLUTION: eSolutions Manual - Powered by Cognero
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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?
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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
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SOLUTION: 10-3 Operations with Radical Expressions
16. SOLUTION: 17. SOLUTION: 18. SOLUTION: 19. SOLUTION: 20. eSolutions
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10-3 Operations with Radical Expressions
20. SOLUTION: 21. SOLUTION: 22. SOLUTION: 23. SOLUTION: eSolutions Manual - Powered by Cognero
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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
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and a length of .
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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.
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10-3 Operations with Radical Expressions
28. SOLUTION: 29. SOLUTION: 30. SOLUTION: eSolutions Manual - Powered by Cognero
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10-3 Operations with Radical Expressions
30. SOLUTION: 31. SOLUTION: 32. SOLUTION: eSolutions Manual - Powered by Cognero
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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.
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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.
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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
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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:
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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
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F 4; 4
G 4; 2
for n = 25.
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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.
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Simplify.
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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
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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
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1
–3
2
≈ –4.2
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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
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5
1
6
≈ 1.4
7
≈ 1.7
8
2
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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
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1
–1
2
≈ –0.6
3
≈ –0.3
4
0
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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
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2
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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
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66. SOLUTION: Page 22
SOLUTION: 10-3 Operations with Radical Expressions
66. SOLUTION: 67. SOLUTION: 68. 3.6t + 6 − 2.5t = 8
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