Download Choose the word or term that best completes each sentence. 1. 7xy

Study Guide and Review - Chapter 7
Choose the word or term that best completes each sentence.
4
1. 7xy is an example of a(n)___________ .
SOLUTION: A product of a number and variables is a monomial.
5
2. The ___________ of 95,234 is 10 .
SOLUTION: 5
5
95,234 is almost 100,000 or 10 , so 10 is its order of magnitude.
3. 2 is a(n) __________ of 8.
SOLUTION: 3
Since 8 = 2 , 2 is the cube root of 8.
4. The rules for operations with exponents can be extended to apply to expressions with a(n) ___________ such as
.
SOLUTION: The exponent is a rational exponent.
n
5. A number written in is of the form a × 10 , where 1 ≤ a < 10 and n is an integer.
SOLUTION: n
A number written as a × 10 , where 1 ≤ a < 10 and n is an integer, is written in scientific notation.
x
6. f (x) = 3 is an example of a(n) ______________ .
SOLUTION: x
A function in the form y = ab , where a ≠ 0, b > 0, and b ≠ 1 is an exponential function.
7. is a(n) ______________ for the sequence .
SOLUTION: A formula that gives the first term of a sequence and tells you how to find the next term when you know the
preceding term is called a recursive formula.
3x – 1
8. 2
= 16 is an example of a(n) _______________ .
SOLUTION: An equation in which the variable occurs as exponent is an exponential equation.
t
9. The equation for _____________ is y = C(1 – r) .
SOLUTION: Since 0 < 1 – r < 1, this is an example of exponential decay.
n
10. If a = b for a positive integer n, then a is a(n) _______________ of b.
SOLUTION: n
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If a Manual
= b for
a positive
integer
Simplify each expression.
3
5
n, then a is an nth root of b.
Page 1
t
9. The equation for _____________ is y = C(1 – r) .
SOLUTION: Study
Guide and Review - Chapter 7
Since 0 < 1 – r < 1, this is an example of exponential decay.
n
10. If a = b for a positive integer n, then a is a(n) _______________ of b.
SOLUTION: n
If a = b for a positive integer n, then a is an nth root of b.
Simplify each expression.
3
11. x ⋅ x ⋅ x
5
SOLUTION: 2 5
12. (2xy)( −3x y )
SOLUTION: 4
5 2
13. (−4ab )(−5a b )
SOLUTION: 3 2 2
14. (6x y )
SOLUTION: 3 3 2
15. [(2r t) ]
SOLUTION: eSolutions Manual - Powered by Cognero
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Study Guide and Review - Chapter 7
3 3 2
15. [(2r t) ]
SOLUTION: 3
16. (−2u )(5u)
SOLUTION: 2 3
3 3
17. (2x ) (x )
SOLUTION: 18. 3 3
(2x )
SOLUTION: eSolutions Manual - Powered by Cognero
2
19. GEOMETRY Use the formula V = πr h to find the volume of the cylinder.
Page 3
Study Guide and Review - Chapter 7
2
19. GEOMETRY Use the formula V = πr h to find the volume of the cylinder.
SOLUTION: Simplify each expression. Assume that no denominator equals zero.
20. SOLUTION: 21. SOLUTION: 22. SOLUTION: eSolutions Manual - Powered by Cognero
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Study Guide and Review - Chapter 7
22. SOLUTION: −3 0 6
23. a b c
SOLUTION: 24. SOLUTION: 25. SOLUTION: eSolutions Manual - Powered by Cognero
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Study Guide and Review - Chapter 7
25. SOLUTION: 26. SOLUTION: 27. SOLUTION: eSolutions Manual - Powered by Cognero
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Study Guide and Review - Chapter 7
27. SOLUTION: 2 4
28. GEOMETRY The area of a rectangle is 25x y square feet. The width of the rectangle is 5xy feet. What is the
length of the rectangle?
SOLUTION: 3
The length of the rectangle is 5xy ft.
Simplify.
29. SOLUTION: 30. SOLUTION: eSolutions Manual - Powered by Cognero
31. Page 7
SOLUTION: Study Guide and Review - Chapter 7
30. SOLUTION: 31. SOLUTION: 32. SOLUTION: 33. SOLUTION: 34. SOLUTION: 35. eSolutions Manual - Powered by Cognero
SOLUTION: Page 8
Study Guide and Review - Chapter 7
35. SOLUTION: 36. SOLUTION: Solve each equation.
x
37. 6 = 7776
SOLUTION: Therefore, the solution is 5.
4x – 1
38. 4
= 32
SOLUTION: Therefore, the solution is .
Express each number in scientific notation.
39. 2,300,000
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SOLUTION: Page 9
Study Guide and Review - Chapter 7
Therefore, the solution is .
Express each number in scientific notation.
39. 2,300,000
SOLUTION: 2,300,000 → 2.300000
The decimal point moved 6 places to the left, so n = 6.
6
2,300,000 = 2.3 × 10
40. 0.0000543
SOLUTION: 0.0000543 → 5.43
The decimal point moved 5 places to the right, so n = –5.
−5
0.0000543 = 5.43 × 10
41. ASTRONOMY Earth has a diameter of about 8000 miles. Jupiter has a diameter of about 88,000 miles. Write in
scientific notation the ratio of Earth’s diameter to Jupiter’s diameter.
SOLUTION: Earth: 8000 = 8.0 × 10
3
Jupiter: 88,000 = 8.8 × 10
4
−2
In scientific notation, the ratio of Earth’s diameter to Jupiter’s diameter is about 9.1 × 10 .
Graph each function. Find the y-intercept, and state the domain and range.
x
42. y = 2
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
y
2
–2
–2
2
–1
2
0
2
1
2
2
2
–1
0
1
1
2
2
4
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Study Guide and Review - Chapter 7
−2
In scientific notation, the ratio of Earth’s diameter to Jupiter’s diameter is about 9.1 × 10 .
Graph each function. Find the y-intercept, and state the domain and range.
x
42. y = 2
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
y
2
–2
–2
2
–1
2
0
2
1
2
2
2
–1
0
1
1
2
2
4
The graph crosses the y-axis at 1. The domain is all real numbers, and the range is all real numbers greater than 0.
x
43. y = 3 + 1
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
y
3 +1
–2
+1
–1
+1
–2
3
–1
3
0
3 +1
1
3 +1
2
3 +1
0
2
1
4
2
10
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The graph crosses the y-axis at 2. The domain is all real numbers, and the range is all real numbers greater than 1.
Study
Guide
Review
- Chapter
The
graphand
crosses
the y-axis
at 1. 7The domain is all real numbers, and the range is all real numbers greater than 0.
x
43. y = 3 + 1
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
y
3 +1
–2
+1
–1
+1
–2
3
–1
3
0
3 +1
1
3 +1
2
3 +1
0
2
1
4
2
10
The graph crosses the y-axis at 2. The domain is all real numbers, and the range is all real numbers greater than 1.
x
44. y = 4 + 2
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
y
4 +2
–2
–2
4
–1
4
0
4 +2
1
4 +2
2
4 +2
–1
+2
+2
0
3
1
6
2
18
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The graph crosses the y-axis at 3. The domain is all real numbers, and the range is all real numbers greater than 2.
x
Study Guide and Review - Chapter 7
The graph crosses the y-axis at 2. The domain is all real numbers, and the range is all real numbers greater than 1.
x
44. y = 4 + 2
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
y
4 +2
–2
–2
4
–1
4
0
4 +2
1
4 +2
2
4 +2
–1
+2
+2
0
3
1
6
2
18
The graph crosses the y-axis at 3. The domain is all real numbers, and the range is all real numbers greater than 2.
x
45. y = 2 − 3
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
2 –3
–2
2
–1
2
0
2 –3
1
2 –3
2
2 –3
–2
–3
–1
–3
y
0
–2
1
–1
2
1
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The graph crosses the y-axis at –2. The domain is all real numbers, and the range is all real numbers greater than –3.
Study Guide and Review - Chapter 7
The graph crosses the y-axis at 3. The domain is all real numbers, and the range is all real numbers greater than 2.
x
45. y = 2 − 3
SOLUTION: Complete a table of values for –2 < x < 2. Connect the points on the graph with a smooth curve. x
x
2 –3
–2
2
–1
2
0
2 –3
1
2 –3
2
2 –3
–2
–3
–1
–3
y
0
–2
1
–1
2
1
The graph crosses the y-axis at –2. The domain is all real numbers, and the range is all real numbers greater than –3.
0.008t
, where t is
46. BIOLOGY The population of bacteria in a petri dish increases according to the model p = 550(2.7)
the number of hours and t = 0 corresponds to 1:00 P.M. Use this model to estimate the number of bacteria in the dish
at 5:00 P.M.
SOLUTION: If t = 0 corresponds to 1:00 P.M, then t = 4 represents 5:00 P.M.
There will be about 568 bacteria in the dish at 5:00 P.M.
47. Find the final value of $2500 invested at an interest rate of 2% compounded monthly for 10 years.
SOLUTION: Use the equation for compound interest, with P = 2500, r = 0.02, n = 12, and t = 10.
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Study Guide and Review - Chapter 7
There will be about 568 bacteria in the dish at 5:00 P.M.
47. Find the final value of $2500 invested at an interest rate of 2% compounded monthly for 10 years.
SOLUTION: Use the equation for compound interest, with P = 2500, r = 0.02, n = 12, and t = 10.
The final value of the investment is about $3053.00.
48. COMPUTERS Zita’s computer is depreciating at a rate of 3% per year. She bought the computer for $1200.
a. Write an equation to represent this situation.
b. What will the computer’s value be after 5 years?
SOLUTION: a. Use the equation for exponential decay, with a = 1200 and r = 0.03.
t
The equation that represents the depreciation of Zita’s computer is y = 1200(1 − 0.03) .
b. Substitute 5 for t and solve.
After 5 years, Zita’s computer value is about $1030.48.
Find the next three terms in each geometric sequence.
49. −1, 1, −1, 1, ...
SOLUTION: Calculate the common ratio.
The common ratio is –1. Multiply each term by the common ratio to find the next three terms.
1 × –1 = –1
–1 × –1 = 1
1 × –1 = –1
The next three terms of the sequence are –1, 1, and –1.
50. 3, 9, 27 ...
SOLUTION: eSolutions
Manual - Powered by Cognero
Calculate the common ratio.
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The common ratio is –1. Multiply each term by the common ratio to find the next three terms.
1 × –1 = –1
–1 × –1 = 1
1 × –1 = –1
Study
Guide
7 are –1, 1, and –1.
The
next and
threeReview
terms of- Chapter
the sequence
50. 3, 9, 27 ...
SOLUTION: Calculate the common ratio.
The common ratio is 3. Multiply each term by the common ratio to find the next three terms.
27 × 3 = 81
81 × 3 = 243
243 × 3 = 729
The next three terms of the sequence are 81, 243, and 729.
51. 256, 128, 64, ...
SOLUTION: Calculate the common ratio.
The common ratio is
. Multiply each term by the common ratio to find the next three terms.
64 × = 32
32 × = 16
16 × = 8
The next three terms of the sequence are 32, 16, and 8.
Write the equation for the nth term of each geometric sequence.
52. −1, 1, −1, 1, ...
SOLUTION: The first term of the sequence is –1. So, a 1 = –1.
Calculate the common ratio.
The common ratio is –1. So, r = –1.
53. 3, 9, 27, ...
SOLUTION: The first term of the sequence is 3. So, a 1 = 3.
Calculate the common ratio.
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The common ratio is –1. So, r = –1.
Study Guide and Review - Chapter 7
53. 3, 9, 27, ...
SOLUTION: The first term of the sequence is 3. So, a 1 = 3.
Calculate the common ratio.
The common ratio is 3. So, r = 3.
54. 256, 128, 64, ...
SOLUTION: The first term of the sequence is 256. So, a 1 = 256.
Calculate the common ratio.
The common ratio is
. So, r =
.
55. SPORTS A basketball is dropped from a height of 20 feet. It bounces to
its height each bounce. Draw a graph to represent the situation.
SOLUTION: Compared to the previous bounce, the ball returns to
its height. So, the common ratio is .
Use common ratio to find next y term
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Study Guide and Review - Chapter 7
Therefore, the geometric sequence that models this situation is 20, 10, 5,
,
,
,
,
, and so forth.
Find the first five terms of each sequence.
56. SOLUTION: Use a 1 = 11 and the recursive formula to find the next four terms.
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Study Guide and Review - Chapter 7
Find the first five terms of each sequence.
56. SOLUTION: Use a 1 = 11 and the recursive formula to find the next four terms.
The first five terms are 11, 7, 3, –1, and –5.
57. SOLUTION: Use a 1 = 3 and the recursive formula to find the next four terms.
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The first five terms are 3, 12, 30, 66, and 138.
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Study
Guide
andterms
Review
Chapter
The
first five
are -11,
7, 3, –1,7 and –5.
57. SOLUTION: Use a 1 = 3 and the recursive formula to find the next four terms.
The first five terms are 3, 12, 30, 66, and 138.
Write a recursive formula for each sequence.
58. 2, 7, 12, 17, ...
SOLUTION: Subtract each term from the term that follows it.
7 – 2 = 5; 12 – 7 = 5, 17 – 12 = 5
There is a common difference of 5. The sequence is arithmetic.
Use the formula for an arithmetic sequence.
The first term a 1 is 2, and n ≥ 2. A recursive formula for the sequence 2, 7, 12, 17, … is a 1 = 2, a n = a n – 1 + 5, n ≥
2.
59. 32, 16, 8, 4, ...
SOLUTION: Subtract each term from the term that follows it.
16 – 32 = –16; 8 – 16 = –8, 4 – 8 = –4
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of 0.5. The sequence is geometric.
Use the formula for a geometric sequence. eSolutions Manual - Powered by Cognero
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The first term a 1 is 2, and n ≥ 2. A recursive formula for the sequence 2, 7, 12, 17, … is a 1 = 2, a n = a n – 1 + 5, n ≥
Study
2. Guide and Review - Chapter 7
59. 32, 16, 8, 4, ...
SOLUTION: Subtract each term from the term that follows it.
16 – 32 = –16; 8 – 16 = –8, 4 – 8 = –4
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is a common ratio of 0.5. The sequence is geometric.
Use the formula for a geometric sequence. The first term a 1 is 32, and n ≥ 2. A recursive formula for the sequence 32, 16, 8, 4, … is a 1 = 32, a n = 0.5a n – 1, n
≥ 2.
60. 2, 5, 11, 23, ...
SOLUTION: Subtract each term from the term that follows it.
5 – 2 = 3; 11 – 5 = 6, 23 – 11 = 12
There is no common difference. Check for a common ratio by dividing each term by the term that precedes it.
There is no common ratio. Therefore the sequence must be a combination of both.
From the difference above, you can see each is twice as big as the previous. So r is 2. From the ratios, if each
denominator was one less, the ratios would be 0.5. Thus, the common difference is 1. Then, if the first term a 1 is 2,
and n ≥ 2, a recursive formula for the sequence 2, 5, 11, 23, … is a 1 = 2, a n = 2a n– 1 + 1, n ≥ 2. eSolutions Manual - Powered by Cognero
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