 4.1: Evaluate each of the following without a calculator. a. 4.2 ? 103 b...
 4.4.1.1: . Write each expression as a power of 10. a. b. c. one billion d. o...
 4.4.2.1: Simplify, when possible, writing your answer as an expression with ...
 4.4.3.1: Simplify and express your answer using positive exponents. Check yo...
 4.4.4.1.: Change the following English units to the metric units indicated. a...
 4.4.5.1: What is the orderofmagnitude difference between the following uni...
 4.4.6.1: Rewrite in an equivalent form using logarithms:
 4.2: Use the rules of exponents to simplify the following. Express your ...
 4.4.1.2: Express in standard decimal notation (without exponents): a.1027 b....
 4.4.2.2: Simplify: b2 (22a4) 2 3 (2a2) 2 3 (1)4 (10a2 b3)3 (2a4)3 (a4)3 (
 4.4.3.2: Simplify and express your answer with positive exponents: a. b. c. ...
 4.4.4.2: Change the following metric units to the English units indicated. a...
 4.4.5.2: Fill in the blanks to make each of the following statements true. a...
 4.4.6.2: Use your calculator to evaluate to two decimal places:
 4.3: For what integer values of x will the following statement be true? ...
 4.4.1.3: Express each in meters, using powers of 10. a. 10 cm c. 3 terameter...
 4.4.2.3: Simplify: (3ab) (22ab 3 1 ab 2)3 (50a10)2 (2x5)3 (24x3)2 1 x3(2x3 2...
 4.4.3.3: Simplify where possible. Express your answer with positive exponents.
 4.4.4.3: For the following questions, make an estimate and then check your e...
 4.4.5.3: Compare the following numbers using orders of magnitude. a. 5.261 a...
 4.4.6.3: Express the number 375 in the form 10x
 4.4: a. Show with an example why the following is not a true statement f...
 4.4.1.4: Express each unit using a metric prefix. a. seconds b. grams c. meters
 4.4.2.4: Simplify and write each variable as an expression with positive exp...
 4.4.3.4: Simplify where possible. Express your answer with positive exponents.
 4.4.4.1.: A football field is 100 yards long. How many meters is this? What p...
 4.4.5.4: An ant is roughly meter in length and the average human roughly one...
 4.4.6.4: Estimate the value of each of the following: a. log 4000 b. log 5,0...
 4.4.6.5: Rewrite the following statements using logs: a. b. c. Rewrite the f...
 4.5: Elephant seals can weigh as much as 5000 lb (for males) and 2000 lb...
 4.4.1.5: Computer storage is often measured in gigabytes and terabytes. Writ...
 4.4.2.5: Simplify and write each variable as an expression with positive exp...
 4.4.3.5: Evaluate without using technology and write the result using scient...
 4.4.4.4: How many droplets of water are in a river that is 100 km long, 250 ...
 4.4.5.5: Refer to the chart on p. 261 in Exploration 4.1. a. How many orders...
 4.4.6.6: Evaluate the following without a calculator. a. Find the following ...
 4.6: An NFL regulation playing field for football is 120 yd (110 m) long...
 4.4.1.6: Express each of the following using powers of 10. a. 10,000,000,000...
 4.4.2.6: Evaluate and express your answer in standard decimal form: a. e. b....
 4.4.3.6: Write each of the following in scientific notation:
 4.4.4.5: a. A roll of aluminum foil claims to be 50 sq ft or 4.65 m2. Show t...
 4.4.5.6: Water boils (changes from a liquid to a gas) at 373 kelvins. The te...
 4.4.6.7: Rewrite the following equations using exponents instead of logs. Es...
 4.7: According to a special report in 2010 by Forbes, the highest paid c...
 4.4.1.7: Write each of the following in scientific notation: a. 0.000 29 d. ...
 4.4.2.7: Convert each number into scientific notation then perform the indic...
 4.4.3.7: Change each number to scientific notation, then simplify using rule...
 4.4.4.1.: If a falling object accelerates at the rate of 9.8 meters per secon...
 4.4.5.7: An electron weighs about gram, and a raindrop weighs about gram. Ho...
 4.4.6.8: Rewrite the following equations using exponents instead of logs. Es...
 4.8: The Yangtze River (China) is 6380 km long. The Colorado River is 14...
 4.4.1.8: Why are the following expressions not in scientific notation? Rewri...
 4.4.2.8: Convert each number into scientific notation and then perform the o...
 4.4.3.8: Use scientific notation and the rules of exponents to perform the i...
 4.4.4.6: Convert the following to feet and express your answers in scientifi...
 4.4.5.8: Determine the orderofmagnitude difference in the sizes of the rad...
 4.4.6.9: . Rewrite the following equations using logs instead of exponents. ...
 4.9: Use the accompanying table to answer the following questions. Count...
 4.4.1.9: Write each of the following in standard decimal form: a. 7.23 ? 105...
 4.4.2.9: Simplify each expression using the properties of exponents.
 4.4.3.9: Evaluate without a calculator:
 4.4.4.7: The speed of light is approximately 1.86 105 miles/sec. a. Write th...
 4.4.5.9: To compare the sizes of different objects, we need to use the same ...
 4.4.6.10: Rewrite the following equations using logs instead of exponents. Es...
 4.10: The Energy Information Administration of the U.S. Department of Ene...
 4.4.1.10: Express each in scientific notation. (Refer to the chart in Explora...
 4.4.2.10: Each of the following simplifications contains an error made by stu...
 4.4.3.10: Evaluate without a calculator:
 4.4.4.1.: . The average distance from Earth to the sun is about 150,000,000 k...
 4.4.5.10: The pH scale measures the hydrogen ion concentration in a liquid, w...
 4.4.6.11: Solve for x. (Hint: Rewrite each expression so that you can use a c...
 4.11: Is the following statement true or false? An increase in one order ...
 4.4.1.11: Determine if the expressions are true or false. If false, change th...
 4.4.2.11: Express your answer as a power of 10 and in standard decimal form. ...
 4.4.3.11: Assume that all variables represent positive quantities and simplify.
 4.4.4.1.: The distance from Earth to the sun is approximately 150 million kil...
 4.4.5.11: Which is an additive scale? Explain why. Which is a multiplicative ...
 4.4.6.12: Without using a calculator, show how you can solve for x.
 4.12: Three projections for the world population (high, medium, and low) ...
 4.4.1.12: Express each quantity in scientific notation. a. The mass of an ele...
 4.4.2.12: In 2010 the Peoples Republic of China was estimated to have about 1...
 4.4.3.12: Simplify by removing all possible factors for each radical. Assume ...
 4.4.4.4: Earth travels in an approximately circular orbit around the sun. Th...
 4.4.5.12: Graph the following on a powerof10 (logarithmic) scale. (See samp...
 4.4.6.13: Without using a calculator show how you can solve for x.
 4.13: Hubbles Law states that galaxies are receding from one another at v...
 4.4.1.13: Evaluate: a. 9 k b. 29 k c.k 21000 k d. 2 k 21000 k
 4.4.2.13: a. In 2010 Japan had a population of approximately 127.4 million pe...
 4.4.3.13: Evaluate each expression without using a calculator.
 4.4.4.5: A barrel of U.S. oil is 42 gallons. A barrel of British oil is 163....
 4.4.5.13: (Refer to the chart on p. 261 in Exploration 4.1.) Describe how to ...
 4.4.6.14: Find the value of x that makes the equation true. a. log x 5 22 b. ...
 4.14: Temperature can affect the speed of sound. The speed of sound, S (i...
 4.4.1.14: The accompanying amusing graph shows a roughly linear relationship ...
 4.4.2.14: The distance that light travels in 1 year (a light year) is miles. ...
 4.4.3.14: Calculate the following:
 4.4.4.1.: A barrel of wheat is 3.2812 bushels (U.S. dry) or 4.0833 cubic feet...
 4.4.5.14: a. Read the chapter entitled The Cosmic Calendar from Carl Sagans b...
 4.4.6.15: Without using a calculator, for each number in the form log x, find...
 4.15: The radius of Earth is about 6.3 106 m and its mass is approximatel...
 4.4.2.15: On April 20th, 2010, an explosion and fire on an oil rig operated b...
 4.4.3.15: Calculate:
 4.4.4.6: In the United States, land is measured in acres and one acre is 43,...
 4.4.5.15: Graph the following on a powerof10 (logarithmic) scale. (See samp...
 4.4.6.16: Use a calculator to determine the following logs. Doublecheck each...
 4.16: Objects that are less dense than water will float; those that are m...
 4.4.2.16: Change each number into scientific notation, then perform the indic...
 4.4.3.16: Evaluate:
 4.4.4.7: Estimate the number of heartbeats in a lifetime. Explain your method.
 4.4.5.16: Radio waves, sent from a broadcast station and picked up by the ant...
 4.4.6.17: On a logarithmic scale, what would correspond to moving over to the...
 4.17: An adult patient weighs 130 lb. The prescription for a drug is 5 mg...
 4.4.2.17: a. For any nonzero real number a, what can we say about the sign of...
 4.4.3.17: Simplify the following expressions by using properties of exponents...
 4.4.4.1.: A nanosecond is 109 second. Modern computers can perform on the ord...
 4.4.6.18: The difference in the noise levels of two sounds is measured in dec...
 4.18: On March 2, 2007, the Boston Globe reported the following: An exaby...
 4.4.2.18: Round off the numbers and then estimate the value of each of the fo...
 4.4.3.18: Each of the following simplifications is false. In each case identi...
 4.4.4.4: Since light takes time to travel, everything we see is from the pas...
 4.4.6.19: The concentration of hydrogen ions in a water solution typically ra...
 4.19: Find the logarithm of each of the following numbers: a. 1 b. 1 bill...
 4.4.2.19: Simplify each expression using two different methods, and then comp...
 4.4.3.19: Evaluate when x 2: a. (x)2 c. x1/2 e. x3/2 b. x2 d. (x)1/2 f. x0 20
 4.4.4.5: Use the conversion factor of 1 light year 9.46 1012 kilometers or 5...
 4.4.6.20: a. Place the number 50 on the additive scale below.
 4.20: Estimate the following by placing the log between the two closest i...
 4.4.2.20: Verify that using the rules of exponents
 4.4.3.20: Determine if the following statements are true or false.
 4.4.4.1.: A homeowner would like to spread shredded bark (mulch) over her flo...
 4.4.6.21: The coordinate system below uses multiplicative or log scales on bo...
 4.21: One way of defining the energy unit joule (J) is the amount of the ...
 4.4.2.21: Verify that using the rules of exponents
 4.4.3.21: Use , , or to make each statement true. a. b. 5 c. "5 ? 2 2 2 42 "3...
 4.4.4.1.: A circular swimming pool is 18 ft in diameter and 4 ft deep. a. Det...
 4.4.6.22: Change each number to a power of 10, then plot the numbers on a pow...
 4.22: Rewrite each number as a power of 10, then create a logarithmic sca...
 4.4.2.22: As of May 22, 2010, the U.S. national debt was estimated at $12,995...
 4.4.3.22: Fill in the missing forms in the table.
 4.4.4.4: An angstrom, , is a metric unit of length equal to one ten billiont...
 4.4.6.23: Compare the times listed below by plotting them on the same ordero...
 4.23: (Requires a calculator that can evaluate powers.) For many drugs, i...
 4.4.2.23: Justify Rule 4 for exponents when n is an integer 0: If a and b are...
 4.4.3.23: A TV signal traveling at the speed of light takes about second to t...
 4.4.4.1.: The National Institutes of Health guidelines suggest that adults ov...
 4.24: Are the statements in 124 true or false? Give an explanation for yo...
 4.4.2.24: Justify Rule 2 for exponents when n m and m and n are integers 0: I...
 4.4.3.24: Estimate the length of a side, s, of a cube with volume, V, of 6 cm...
 4.4.4.4: Computer technology refers to the storage capacity for information ...
 4.25: In 2530, give examples with the specified properties Populations of...
 4.4.2.25: In 2008 the United Kingdom generated approximately 53 terawatthour...
 4.4.3.25: Estimate the radius of a spherical balloon that has a volume of 4 f...
 4.4.4.5: If 1 angstrom, , 1010 meter, determine the following values. a. The...
 4.26: In 2530, give examples with the specified properties A number x suc...
 4.4.3.26: Constellation. Reduce each of the following expressions to the form...
 4.4.4.1.: Anthrax spores, which were inhaled by postal workers, causing sever...
 4.27: In 2530, give examples with the specified properties A number x suc...
 4.4.3.27: An equilateral triangle has sides of length 8 cm. a. Find the heigh...
 4.28: In 2530, give examples with the specified properties A positive num...
 4.4.3.28: An Egyptian pyramid consists of a square base and four triangular s...
 4.29: In 2530, give examples with the specified properties A nonzero num...
 4.4.3.29: Simplify each expression by removing all possible factors from the ...
 4.30: In 2530, give examples with the specified properties A number b suc...
 4.4.3.30: (Requires a calculator that can evaluate powers.) Perform the follo...
 4.31: Are the statements in 3140 true or false If one quantity is four or...
 4.4.3.31: Describe at least three different methods for entering into a calcu...
 4.32: Are the statements in 3140 true or false k c k 5 c for any real num...
 4.4.3.32: Using rules of exponents, show that 95 2727 5 331
 4.33: Are the statements in 3140 true or false og x is defined only for n...
 4.4.3.33: Using rules of exponents, show that 1 x2n 5 xn
 4.34: Are the statements in 3140 true or false Raising a number to the po...
 4.4.3.34: Write an expression that shows the calculation(s) necessary to answ...
 4.35: Are the statements in 3140 true or false bm ? bm 5 bm2
 4.4.3.35: The time it takes for one complete swing of a pendulum is called th...
 4.36: Are the statements in 3140 true or false (bp)q 5 bp1q
 4.4.3.36: (Requires the use of a calculator that can evaluate powers.) A whee...
 4.37: Are the statements in 3140 true or false (b 1 c)m 5 bm 1 cm
 4.4.3.37: (Requires the use of a calculator that can evaluate powers.) The br...
 4.38: Are the statements in 3140 true or false (2b)q 5 bq
 4.4.3.38: Requires the use of a calculator that can evaluate powers.) If a ro...
 4.39: Are the statements in 3140 true or falsebm ? cn 5 (b ? c)m1n
 4.40: Are the statements in 3140 true or false If n is odd, can be positi...
Solutions for Chapter 4: THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE
Full solutions for Explorations in College Algebra  5th Edition
ISBN: 9780470466445
Solutions for Chapter 4: THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE
Get Full SolutionsSince 182 problems in chapter 4: THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE have been answered, more than 8215 students have viewed full stepbystep solutions from this chapter. Chapter 4: THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE includes 182 full stepbystep solutions. This textbook survival guide was created for the textbook: Explorations in College Algebra, edition: 5. Explorations in College Algebra was written by and is associated to the ISBN: 9780470466445. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.