Download Simplify each expression. 1. SOLUTION: 2. SOLUTION: 3

10-3 Operations with Radical Expressions
Simplify each expression.
7. 1. SOLUTION: SOLUTION: 2. SOLUTION: 8. SOLUTION: 3. SOLUTION: 9. SOLUTION: 4. SOLUTION: 5. SOLUTION: 10. SOLUTION: 6. SOLUTION: 11. SOLUTION: 7. 12. SOLUTION: eSolutions Manual - Powered by Cognero
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SOLUTION: SOLUTION: 10-3 Operations with Radical Expressions
16. 12. SOLUTION: 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?
17. SOLUTION: SOLUTION: 18. SOLUTION: 19. SOLUTION: The area of the triangle is
Simplify each expression.
.
20. 14. SOLUTION: SOLUTION: 15. SOLUTION: 21. SOLUTION: eSolutions Manual - Powered by Cognero
16. SOLUTION: Page 2
10-3 Operations with Radical Expressions
26. GEOMETRY Find the perimeter and area of a
21. and a length of
rectangle with a width of
SOLUTION: .
SOLUTION: 22. SOLUTION: The perimeter is
units.
23. SOLUTION: The area is 12 square units.
Simplify each expression.
24. 27. SOLUTION: SOLUTION: 25. SOLUTION: 28. SOLUTION: 26. GEOMETRY Find the perimeter and area of a
rectangle with a width of
and a length of
.
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29. 10-3 Operations with Radical Expressions
32. 28. SOLUTION: 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
29. .
SOLUTION: 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.
30. SOLUTION: The coaster must have a velocity of 0 feet per
second at the top of the hill.
31. SOLUTION: 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
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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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second at the top of the hill.
b. Sample answer: In the formula given, we are
the square
of the Expressions
difference, not the
10-3taking
Operations
withroot
Radical
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
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
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.
, 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.
The average annual interest rate that Tadi’s account
earned was about 1.5%.
There are
or about 13 amps of electrical current running through the microwave oven.
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
36. CHALLENGE Determine whether the following
statement is true or false . Provide a proof or
counterexample to support your answer.
x +y >
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.
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.
There are
or about 13 amps of electrical current running through the microwave oven.
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36. CHALLENGE Determine whether the following
statement is true or false . Provide a proof or
Therefore,
for all x > 0 and y > 0.
is a true statement
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37. CCSS ARGUMENTS Make a conjecture about
are
or about 13 amps of electrical 10-3There
Operations
with
Radical Expressions
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.
when x > 0 and y > 0
x +y >
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.
Therefore,
for all x > 0 and y > 0.
is a true statement
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. Because x > 0 and y > 0, the product 2xy must be
a positive number.Thus, 2xy > 0 is always true.
Therefore,
for all x > 0 and y > 0.
is a true statement
is irrational.
Examine the product of several pairs of non-zero
rational and irrational numbers:
is irrational
is irrational
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. 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:
is irrational.
Examine the product of several pairs of non-zero
rational and irrational numbers:
is irrational
When you simplify
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is irrational
is irrational. , you get
. When you
simplify
, you get
. Because these two
numbers have the same radicand, you can add them.
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39. WRITING IN MATH Describe step by step how
to multiply two radical expressions, each with two
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. 10-3 Operations with Radical Expressions
38. OPEN ENDED Write an equation that shows a
sum of two radicals with different radicands. Explain
how you could combine these terms.
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?
SOLUTION: Sample answer:
A 2(a + b + c)
B 3(a + b + c)
When you simplify
, you get
. When you
simplify
, you get
. Because these two
numbers have the same radicand, you can add them.
C 4(a + b + c)
D 12(a + b + c)
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.
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.
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
In 6 years, the population of the town will be 14,500.
Which
expression
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represents the
sum of the lengths of the 12 edges on this rectangular
solid?
, 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.
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10-3 Operations with Radical Expressions
Choice G is correct. 43. The current I in a simple electrical circuit is given by
the formula
45. , where V is the voltage and R is
SOLUTION: 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.
46. C The current will be half its previous value.
SOLUTION: 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. 47. Consider an example with the resistance is 4 and
voltage is 100. Find I when R doubles. SOLUTION: 48. SOLUTION: Choice C is the correct answer.
49. SOLUTION: Simplify.
44. SOLUTION: Graph each function. Compare to the parent
graph. State the domain and range.
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45. SOLUTION: SOLUTION: x
0
y
0
0.5
≈ 1.4
1
2
2
≈ 2.8
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and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –3 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
SOLUTION: 10-3 Operations with Radical Expressions
Graph each function. Compare to the parent
graph. State the domain and range.
52. SOLUTION: x
–1
–0.5
y
0
≈ 0.7
50. SOLUTION: x
0
y
0
0.5
≈ 1.4
1
2
0
1
1
≈ 1.4
2
≈ 1.7
3
≈ 2
2
≈ 2.8
The value 1 is being added to the square root of the
parent function
, so the graph is translated 1
is multiplied by a value
The parent function greater than 1, so the graph is a vertical stretch of
. 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}.
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 xvalues for the parent function. The domain is {x|x ≥ –1}, and the range is {y|y ≥ 0}.
53. 51. SOLUTION: x
0
y
0
0.5
≈ –2.1
1
–3
SOLUTION: x
4
4.5
y
0
≈ 0.7
2
≈ –4.2
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}.
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 yvalues in the table are –3 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
54. 52. SOLUTION: eSolutions Manual - Powered by Cognero
x
0
1
–1
–0.5
y
0
1
≈ 0.7
≈ 1.4
2
≈ 1.7
3
≈ 2
SOLUTION: x
0
0.5
y
3
≈ 3.7
1
4
2
≈ 4.4
3
≈ 4.7
4
5
Page 9
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
10-3corresponding
Operations with
Radical
Expressions
x-values
for the
parent function. The
domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}.
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
54. SOLUTION: x
0
0.5
y
3
≈ 3.7
1
4
2
≈ 4.4
3
≈ 4.7
4
5
SOLUTION: 2
57. y + 13y + 30
SOLUTION: 2
The value 3 is being added to the parent function
, 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}.
58. p − 17p + 72
SOLUTION: 2
59. x + 6x – 7
SOLUTION: 55. SOLUTION: x
0
0.5
y
–2
≈ –
1.3
1
–1
2
≈ –
0.6
3
≈ –
0.3
4
0
2
60. y − y − 42
SOLUTION: 2
61. −72 + 6w + w
SOLUTION: The value 2 is being subtracted from the parent
function
, 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}.
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.
Factor each trinomial.
2
56. x + 12x + 27
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SOLUTION: 10-3 Operations with Radical Expressions
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.
67. SOLUTION: 68. 3.6t + 6 − 2.5t = 8
The value of the investment after 7 years is about
$661.44.
SOLUTION: 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: 66. SOLUTION: 67. eSolutions Manual - Powered by Cognero
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