Choose the word or term that best completes each sentence.
1.7xy4 is an example of a(n)___________ .
SOLUTION:
A product of a number and variables is a monomial.
2.The ___________ of 95,234 is 105.
SOLUTION:
95,234 is almost 100,000 or 105, so 105 is its order of magnitude.
3.2 is a(n) __________ of 8.
SOLUTION:
Since 8 = 23, 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 isarationalexponent.
5.A number written in is of the form a×10n,where1≤a < 10 and n is an integer.
SOLUTION:
A number written as a×10n,where1≤a < 10 and n is an integer, is written in scientific notation.
6.f(x) = 3x is an example of a(n) ______________ .
SOLUTION:
A function in the form y = abx, where a≠0,b > 0, and b≠1isanexponentialfunction.
7. isa(n)______________forthesequence .
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.
8.23x – 1 = 16 is an example of a(n) _______________ .
SOLUTION:
An equation in which the variable occurs as exponent is an exponential equation.
9.The equation for _____________ is y = C(1 – r)t .
SOLUTION:
Since 0 < 1 – r < 1, this is an example of exponential decay.
10.If an = b for a positive integer n, then a is a(n) _______________ of b.
SOLUTION:
If an = b for a positive integer n, then a is an nth root of b.
Simplify each expression.
11.x ⋅ x3 ⋅ x5
SOLUTION:
12.(2xy)( −3x2y5)
SOLUTION:
13.(−4ab4)(−5a5b2)
SOLUTION:
14.(6x3y2)2
SOLUTION:
15.[(2r3t)3]2
SOLUTION:
16.(−2u3)(5u)
SOLUTION:
17.(2x2)3(x3)3
SOLUTION:
18. (2x3)3
SOLUTION:
19.GEOMETRY Use the formula V = πr2h to find the volume of the cylinder.
SOLUTION:
Simplify each expression. Assume that no denominator equals zero.
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.a−3b0c6
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.GEOMETRY The area of a rectangle is 25x2y4 square feet. The width of the rectangle is 5xy feet. What is the
length of the rectangle?
SOLUTION:
The length of the rectangle is 5xy3 ft.
Simplify.
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35.
SOLUTION:
36.
SOLUTION:
Solve each equation.
37.6x = 7776
SOLUTION:
Therefore, the solution is 5.
38.44x – 1 = 32
SOLUTION:
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.
2,300,000=2.3×106
40.0.0000543
SOLUTION:
0.0000543→5.43
The decimal point moved 5 places to the right, so n = –5.
0.0000543=5.43×10−5
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=8.0×103
Jupiter:88,000=8.8×104
In scientific notation, the ratio of Earth’s diameter to Jupiter’sdiameterisabout9.1×10−2.
Graph each function. Find the y-intercept, and state the domain and range.
42.y = 2x
SOLUTION:
Complete a table of values for –2 < x<2.Connectthepointsonthegraphwithasmoothcurve.
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 2x y
–2 2–2
–1 2–1
0 20 1
1 21 2
2 22 4
43.y = 3x + 1
SOLUTION:
Complete a table of values for –2 < x<2.Connectthepointsonthegraphwithasmoothcurve.
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 3x + 1 y
–2 3–2 + 1
–1 3–1 + 1
0 30 + 1 2
1 31 + 1 4
2 32 + 1 10
44.y = 4x + 2
SOLUTION:
Complete a table of values for –2 < x<2.Connectthepointsonthegraphwithasmoothcurve.
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 4x + 2 y
–2 4–2 + 2
–1 4–1 + 2
0 40 + 2 3
1 41 + 2 6
2 42 + 2 18
45.y = 2x − 3
SOLUTION:
Complete a table of values for –2 < x<2.Connectthepointsonthegraphwithasmoothcurve.
The graph crosses the y-axis at –2. The domain is all real numbers, and the range is all real numbers greater than –3.
x 2x – 3 y
–2 2–2 – 3
–1 2–1 – 3
0 20 – 3 –2
1 21 – 3 –1
2 22 – 3 1
46.BIOLOGY The population of bacteria in a petri dish increases according to the model p = 550(2.7)0.008t, where t is
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.
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.
The equation that represents the depreciation of Zita’s computer is y = 1200(1 − 0.03)t.
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:
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, a1 = –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, a1 = 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, a1 = 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 itsheighteachbounce.Drawagraph
to represent the situation.
SOLUTION:
Compared to the previous bounce, the ball returns to itsheight.So,thecommonratiois .
Use common ratio to find next y term
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 a1 = 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 a1 = 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 a1 is 2, and n≥ 2. A recursive formula for the sequence 2, 7, 12, 17, …is a1 = 2, an = an – 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.
Usetheformulaforageometricsequence.
The first term a1 is 32, and n≥2.Arecursiveformulaforthesequence32,16,8,4,…is a1 = 32, an = 0.5an – 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 a1 is 2,
and n≥ 2, a recursive formula for the sequence 2, 5, 11, 23, …is a1 = 2, an = 2an– 1 + 1, n≥ 2.
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Study Guide and Review - Chapter 7
