3.7: Exercise Supplement
- Page ID
- 52132
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exponents and Roots
For problems 1 -25, determine the value of each power and root.
Exercise \(\PageIndex{1}\)
\(3^3\)
- Answer
-
27
Exercise \(\PageIndex{2}\)
\(4^3\)
Exercise \(\PageIndex{3}\)
\(0^5\)
- Answer
-
0
Exercise \(\PageIndex{4}\)
\(1^4\)
Exercise \(\PageIndex{5}\)
\(12^2\)
- Answer
-
144
Exercise \(\PageIndex{6}\)
\(7^2\)
Exercise \(\PageIndex{7}\)
\(8^2\)
- Answer
-
64
Exercise \(\PageIndex{8}\)
\(11^2\)
Exercise \(\PageIndex{9}\)
\(2^5\)
- Answer
-
32
Exercise \(\PageIndex{10}\)
\(3^4\)
Exercise \(\PageIndex{11}\)
\(15^2\)
- Answer
-
225
Exercise \(\PageIndex{12}\)
\(20^2\)
Exercise \(\PageIndex{13}\)
\(25^2\)
- Answer
-
625
Exercise \(\PageIndex{14}\)
\(\sqrt{36}\)
Exercise \(\PageIndex{15}\)
\(\sqrt{225}\)
- Answer
-
15
Exercise \(\PageIndex{16}\)
\(\sqrt[3]{64}\)
Exercise \(\PageIndex{17}\)
\(\sqrt[4]{16}\)
- Answer
-
2
Exercise \(\PageIndex{18}\)
\(\sqrt{0}\)
Exercise \(\PageIndex{19}\)
\(\sqrt[3]{1}\)
- Answer
-
1
Exercise \(\PageIndex{20}\)
\(\sqrt[3]{216}\)
Exercise \(\PageIndex{21}\)
\(\sqrt{144}\)
- Answer
-
12
Exercise \(\PageIndex{22}\)
\(\sqrt{196}\)
Exercise \(\PageIndex{23}\)
\(\sqrt{1}\)
- Answer
-
1
Exercise \(\PageIndex{24}\)
\(\sqrt[4]{0}\)
Exercise \(\PageIndex{25}\)
\(\sqrt[6]{64}\)
- Answer
-
2
Section 3.2
For problems 26-45, use the order of operations to determine each value.
Exercise \(\PageIndex{26}\)
\(2^3 - 2 \cdot 4\)
Exercise \(\PageIndex{27}\)
\(5^2 - 10 \cdot 2 - 5\)
- Answer
-
0
Exercise \(\PageIndex{28}\)
\(\sqrt{81} - 3^2 + 6 \cdot 2\)
Exercise \(\PageIndex{29}\)
\(15^2 + 5^2 \cdot 2^2\)
- Answer
-
325
Exercise \(\PageIndex{30}\)
\(3 \cdot (2^2 + 3^2)\)
Exercise \(\PageIndex{31}\)
\(64 \cdot (3^2 - 2^3)\)
- Answer
-
64
Exercise \(\PageIndex{32}\)
\(\dfrac{5^2 + 1}{13} + \dfrac{3^3 + 1}{14}\)
Exercise \(\PageIndex{33}\)
\(\dfrac{6^2 - 1}{5 \cdot 7} - \dfrac{49 + 7}{2 \cdot 7}\)
- Answer
-
-3
Exercise \(\PageIndex{34}\)
\(\dfrac{2 \cdot [3 + 5(2^2 + 1)]}{5 \cdot 2^3 - 3^2}\)
Exercise \(\PageIndex{35}\)
\(\dfrac{3^2 \cdot [2^5 - 1^4 (2^3 + 25)]}{2 \cdot 5^2 + 5 + 2}\)
- Answer
-
\(-\dfrac{9}{57}\)
Exercise \(\PageIndex{36}\)
\(\dfrac{(5^2 - 2^3) - 2 \cdot 7}{2^2 - 1} + 5 \cdot [\dfrac{3^2 - 3}{2} + 1]\)
Exercise \(\PageIndex{37}\)
\((8 - 3)^2 + (2 + 3^2)^2\)
- Answer
-
146
Exercise \(\PageIndex{38}\)
\(3^2 \cdot (4^2 + \sqrt{25}) + 2^3 \cdot (\sqrt{81} - 3^2)\)
Exercise \(\PageIndex{39}\)
\(\sqrt{16 + 9}}\)
- Answer
-
5
Exercise \(\PageIndex{40}\)
\(\sqrt{16} + \sqrt{9}\)
Exercise \(\PageIndex{41}\)
Compare the results of problems 39 and 40. What might we conclude?
- Answer
-
The sum of square roots is not necessarily equal to the square root of the sum.
Exercise \(\PageIndex{42}\)
\(\sqrt{18 \cdot 2}\)
Exercise \(\PageIndex{43}\)
\(\sqrt{6 \cdot 6}\)
- Answer
-
6
Exercise \(\PageIndex{44}\)
\(\sqrt{7 \cdot 7}\)
Exercise \(\PageIndex{45}\)
\(\sqrt{8 \cdot 8}\)
- Answer
-
8
Exercise \(\PageIndex{46}\)
An records the number of identical factors that are repeated in a multiplication.
Prime Factorization of Natural Numbers
For problems 47- 53, find all the factors of each number.
Exercise \(\PageIndex{47}\)
18
- Answer
-
1, 2, 3, 6, 9, 18
Exercise \(\PageIndex{48}\)
24
Exercise \(\PageIndex{49}\)
11
- Answer
-
1, 11
Exercise \(\PageIndex{50}\)
12
Exercise \(\PageIndex{51}\)
51
- Answer
-
1, 3, 17, 51,
Exercise \(\PageIndex{52}\)
25
Exercise \(\PageIndex{53}\)
2
- Answer
-
1, 2
Exercise \(\PageIndex{54}\)
What number is the smallest prime number?
Grouping Symbol and the Order of Operations
For problems 55 -64, write each number as a product of prime factors.
Exercise \(\PageIndex{55}\)
55
- Answer
-
\(5 \cdot 11\)
Exercise \(\PageIndex{56}\)
20
Exercise \(\PageIndex{57}\)
80
- Answer
-
\(2^4 \cdot 5\)
Exercise \(\PageIndex{58}\)
284
Exercise \(\PageIndex{59}\)
700
- Answer
-
\(2^2 \cdot 5^2 \cdot 7\)
Exercise \(\PageIndex{60}\)
845
Exercise \(\PageIndex{61}\)
1,614
- Answer
-
\(2 \cdot 3 \cdot 269\)
Exercise \(\PageIndex{62}\)
921
Exercise \(\PageIndex{63}\)
29
- Answer
-
29 is a prime number
Exercise \(\PageIndex{64}\)
37
The Greatest Common Factor
For problems 65 - 75, find the greatest common factor of each collection of numbers.
Exercise \(\PageIndex{65}\)
5 and 15
- Answer
-
5
Exercise \(\PageIndex{66}\)
6 and 14
Exercise \(\PageIndex{67}\)
10 and 15
- Answer
-
5
Exercise \(\PageIndex{68}\)
6, 8, and 12
Exercise \(\PageIndex{69}\)
18 and 24
- Answer
-
6
Exercise \(\PageIndex{70}\)
42 and 54
Exercise \(\PageIndex{71}\)
40 and 60
- Answer
-
20
Exercise \(\PageIndex{72}\)
18, 48, and 72
Exercise \(\PageIndex{73}\)
147, 189, and 315
- Answer
-
21
Exercise \(\PageIndex{74}\)
64, 72, and 108
Exercise \(\PageIndex{75}\)
275, 297, and 539
- Answer
-
11
The Least Common Multiple
For problems 76-86, find the least common multiple of each collection of numbers.
Exercise \(\PageIndex{76}\)
5 and 15
Exercise \(\PageIndex{77}\)
6 and 14
- Answer
-
42
Exercise \(\PageIndex{78}\)
10 and 15
Exercise \(\PageIndex{79}\)
36 and 90
- Answer
-
180
Exercise \(\PageIndex{80}\)
42 and 54
Exercise \(\PageIndex{81}\)
8, 12, and 20
- Answer
-
120
Exercise \(\PageIndex{82}\)
40, 50, and 180
Exercise \(\PageIndex{83}\)
135, 147, and 324
- Answer
-
79, 380
Exercise \(\PageIndex{84}\)
108, 144, and 324
Exercise \(\PageIndex{85}\)
5, 18, 25, and 30
- Answer
-
450
Exercise \(\PageIndex{86}\)
12, 15, 18, and 20
Exercise \(\PageIndex{87}\)
Find all divisors of 24.
- Answer
-
1, 2, 3, 4, 6, 8, 12, 24
Exercise \(\PageIndex{88}\)
Find all factors of 24.
Exercise \(\PageIndex{89}\)
Write all divisors of \(2^3 \cdot 5^2 \cdot 7\).
- Answer
-
1, 2, 4, 5, 7, 8, 10, 14, 20, 25, 35, 40, 50, 56, 70, 100, 140, 175, 200, 280, 700, 1,400
Exercise \(\PageIndex{90}\)
Write all divisors of \(6 \cdot 8^2 \cdot 10^3\).
Exercise \(\PageIndex{91}\)
Does 7 divide \(5^3 \cdot 6^4 \cdot 7^2 \cdot 8^5\)?
- Answer
-
yes
Exercise \(\PageIndex{86}\)
Does 13 divide \(8^3 \cdot 10^2 \cdot 11^4 \cdot 13^2 \cdot 15\)?