 5.6.5.1.487: In the expression 23 , the 3 is known as the .
 5.6.5.1.488: In the expression 23 , the 2 is known as the .
 5.6.5.1.489: The product x # x # x # x # x can also be written as the exponentia...
 5.6.5.1.490: With the product rule for exponents, the expression x2 # x3 can be ...
 5.6.5.1.491: With the quotient rule for exponents, the expression x7 x4 can be s...
 5.6.5.1.492: With the zero exponent rule, the expression 70 can be simplified to .
 5.6.5.1.493: With the negative exponent rule, for x 0, the expression x5 can be...
 5.6.5.1.494: With the power rule for exponents, the expression (x2 ) 3 can be si...
 5.6.5.1.495: When written in scientific notation, the number 2013 is .
 5.6.5.1.496: When written in scientific notation, the number 0.0034 is .
 5.6.5.1.497: When written in decimal notation, the number 5.73 * 105 is .
 5.6.5.1.498: When written in decimal notation, the number 1.776 * 103 is .
 5.6.5.1.499: In Exercises 13 40, evaluate the expression. Assume x 0.a) 32 b) 23
 5.6.5.1.500: In Exercises 13 40, evaluate the expression. Assume x 0.a) 25 b) 52
 5.6.5.1.501: In Exercises 13 40, evaluate the expression. Assume x 0.a) (5)2 b)...
 5.6.5.1.502: In Exercises 13 40, evaluate the expression. Assume x 0.a) 32 b) (...
 5.6.5.1.503: In Exercises 13 40, evaluate the expression. Assume x 0.a) 24 b) (...
 5.6.5.1.504: In Exercises 13 40, evaluate the expression. Assume x 0.a) (3)4 b)...
 5.6.5.1.505: In Exercises 13 40, evaluate the expression. Assume x 0.a) 43 b) (...
 5.6.5.1.506: In Exercises 13 40, evaluate the expression. Assume x 0.a) (2)7 b)...
 5.6.5.1.507: In Exercises 13 40, evaluate the expression. Assume x 0.a)  a 1 2 ...
 5.6.5.1.508: In Exercises 13 40, evaluate the expression. Assume x 0.a)  a 1 2 ...
 5.6.5.1.509: In Exercises 13 40, evaluate the expression. Assume x 0.a) 1001 b) ...
 5.6.5.1.510: In Exercises 13 40, evaluate the expression. Assume x 0.a) 20121 b)...
 5.6.5.1.511: In Exercises 13 40, evaluate the expression. Assume x 0.a) 32 # 33 ...
 5.6.5.1.512: In Exercises 13 40, evaluate the expression. Assume x 0.a) 23 # 2 b...
 5.6.5.1.513: In Exercises 13 40, evaluate the expression. Assume x 0.a) 57 55 b)...
 5.6.5.1.514: In Exercises 13 40, evaluate the expression. Assume x 0.a) 45 42 b)...
 5.6.5.1.515: In Exercises 13 40, evaluate the expression. Assume x 0.a) 60 b) 60
 5.6.5.1.516: In Exercises 13 40, evaluate the expression. Assume x 0.a) (6)0 b)...
 5.6.5.1.517: In Exercises 13 40, evaluate the expression. Assume x 0.a) (6x) 0 b...
 5.6.5.1.518: In Exercises 13 40, evaluate the expression. Assume x 0.a) 6x0 b) ...
 5.6.5.1.519: In Exercises 13 40, evaluate the expression. Assume x 0.a) 33 b) 72
 5.6.5.1.520: In Exercises 13 40, evaluate the expression. Assume x 0.a) 51 b) 24
 5.6.5.1.521: In Exercises 13 40, evaluate the expression. Assume x 0.a) 92 b) ...
 5.6.5.1.522: In Exercises 13 40, evaluate the expression. Assume x 0.a) 52 b) ...
 5.6.5.1.523: In Exercises 13 40, evaluate the expression. Assume x 0.a) (22 ) 3 ...
 5.6.5.1.524: In Exercises 13 40, evaluate the expression. Assume x 0.a) c a 1 2 ...
 5.6.5.1.525: In Exercises 13 40, evaluate the expression. Assume x 0.a) 43 # 42...
 5.6.5.1.526: In Exercises 13 40, evaluate the expression. Assume x 0.a) (5)2 (...
 5.6.5.1.527: In Exercises 4152, express the number in scientific notation.415000
 5.6.5.1.528: In Exercises 4152, express the number in scientific notation.923000000
 5.6.5.1.529: In Exercises 4152, express the number in scientific notation.0.00275
 5.6.5.1.530: In Exercises 4152, express the number in scientific notation.0.000034
 5.6.5.1.531: In Exercises 4152, express the number in scientific notation.0.56
 5.6.5.1.532: In Exercises 4152, express the number in scientific notation.0.00467
 5.6.5.1.533: In Exercises 4152, express the number in scientific notation.19000
 5.6.5.1.534: In Exercises 4152, express the number in scientific notation.126000...
 5.6.5.1.535: In Exercises 4152, express the number in scientific notation.0.000186
 5.6.5.1.536: In Exercises 4152, express the number in scientific notation.0.0003
 5.6.5.1.537: In Exercises 4152, express the number in scientific notation.0.0000...
 5.6.5.1.538: In Exercises 4152, express the number in scientific notation.54000
 5.6.5.1.539: In Exercises 5362, express the number in decimal notation. 4.2 * 104
 5.6.5.1.540: In Exercises 5362, express the number in decimal notation. 3.9 * 105
 5.6.5.1.541: In Exercises 5362, express the number in decimal notation. 1.32 * 102
 5.6.5.1.542: In Exercises 5362, express the number in decimal notation. 4.003 * 107
 5.6.5.1.543: In Exercises 5362, express the number in decimal notation. 8.62 * 105
 5.6.5.1.544: In Exercises 5362, express the number in decimal notation. 2.19 * 104
 5.6.5.1.545: In Exercises 5362, express the number in decimal notation. 3.12 * 101
 5.6.5.1.546: In Exercises 5362, express the number in decimal notation. 4.6 * 101
 5.6.5.1.547: In Exercises 5362, express the number in decimal notation. 9 * 106
 5.6.5.1.548: In Exercises 5362, express the number in decimal notation. 7.3 * 104
 5.6.5.1.549: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.550: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.551: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.552: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.553: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.554: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.555: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.556: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.557: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.558: In Exercises 6372, (a) perform the indicated operation without the ...
 5.6.5.1.559: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.560: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.561: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.562: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.563: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.564: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.565: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.566: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.567: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.568: In Exercises 73 82, (a) perform the indicated operation without the...
 5.6.5.1.569: In Exercises 8386, list the numbers from smallest to largest. 1.03 ...
 5.6.5.1.570: In Exercises 8386, list the numbers from smallest to largest.8.5 * ...
 5.6.5.1.571: In Exercises 8386, list the numbers from smallest to largest.40,000...
 5.6.5.1.572: In Exercises 8386, list the numbers from smallest to largest.267,00...
 5.6.5.1.573: U.S. Debt per Person: 2006 Versus 2010 In Example 14 on page 265, t...
 5.6.5.1.574: United Kingdom Debt per Person In 2010, the United Kingdoms governm...
 5.6.5.1.575: U.S. Population In June 2010, the population of the United States w...
 5.6.5.1.576: Chinas Population In June 2010, the population of China was about 1...
 5.6.5.1.577: New York Population Density In June 2010, the population of the met...
 5.6.5.1.578: Mumbai Population Density In June 2010, the population of the metro...
 5.6.5.1.579: Gross Domestic Product The gross domestic product (GDP) of a countr...
 5.6.5.1.580: Chinas GDP In 2009, the GDP (see Exercise 93) of China was about $4...
 5.6.5.1.581: Traveling to the Moon The distance from Earth to the moon is approx...
 5.6.5.1.582: Traveling to Jupiter The distance from Earth to the planet Jupiter ...
 5.6.5.1.583: Traveling to Proxima Centauri The star closest in distance to our o...
 5.6.5.1.584: Computer Speed As of June 22, 2010, the fastest computer in the wor...
 5.6.5.1.585: 1950 Niagara Treaty The 1950 Niagara Treaty between the United Stat...
 5.6.5.1.586: Bucket Full of Molecules A drop of water contains about 40 billion ...
 5.6.5.1.587: Blood Cells in a Cubic Millimeter If a cubic millimeter of blood co...
 5.6.5.1.588: Radioactive Isotopes The halflife of a radioactive isotope is the ...
 5.6.5.1.589: Mutual Fund Manager Lauri Mackey is the fund manager for the Mackey...
 5.6.5.1.590: Another Mutual Fund Manager Susan Dratch is the fund manager for th...
 5.6.5.1.591: Metric System Comparison In the metric system, 1 meter = 103 millim...
 5.6.5.1.592: Milligrams and Kilograms In the metric system, 1 gram = 103 milligr...
 5.6.5.1.593: Earth to Sun Comparison The mass of the sun is approximately 2 * 10...
 5.6.5.1.594: Comparing a Million to a Billion Many people have no idea of the di...
 5.6.5.1.595: Speed of Light a) Light travels at a speed of 1.86 * 105 mi/sec. A ...
 5.6.5.1.596: Bacteria in a Culture The exponential function E(t) = 210 # 2t appr...
 5.6.5.1.597: Obtain data from the U.S. Department of the Treasury and from the U...
 5.6.5.1.598: John Allen Paulos Read the Profiles in Mathematics box on page 264....
 5.6.5.1.599: Find an article in a newspaper or magazine that contains scientific...
Solutions for Chapter 5.6: Number Theory and the Real Number System
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 5.6: Number Theory and the Real Number System
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. Chapter 5.6: Number Theory and the Real Number System includes 113 full stepbystep solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Since 113 problems in chapter 5.6: Number Theory and the Real Number System have been answered, more than 148969 students have viewed full stepbystep solutions from this chapter.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.