 1.3.1.3.1: Add or subtract as indicated. See Examples 1 through 33 + 8
 1.3.1.3.2: Add or subtract as indicated. See Examples 1 through 312 + 1 72
 1.3.1.3.3: Add or subtract as indicated. See Examples 1 through 314 + 1 102
 1.3.1.3.4: Add or subtract as indicated. See Examples 1 through 35 + 1 92
 1.3.1.3.5: Add or subtract as indicated. See Examples 1 through 34.3  6.7
 1.3.1.3.6: Add or subtract as indicated. See Examples 1 through 38.2  1 6.62
 1.3.1.3.7: Add or subtract as indicated. See Examples 1 through 313  17
 1.3.1.3.8: Add or subtract as indicated. See Examples 1 through 315  1 12 16
 1.3.1.3.9: Add or subtract as indicated. See Examples 1 through 31115  a  35b
 1.3.1.3.10: Add or subtract as indicated. See Examples 1 through 3710  45
 1.3.1.3.11: Add or subtract as indicated. See Examples 1 through 319  10  11
 1.3.1.3.12: Add or subtract as indicated. See Examples 1 through 313  4 + 9
 1.3.1.3.13: Add or subtract as indicated. See Examples 1 through 3 45  a  310 b
 1.3.1.3.14: Add or subtract as indicated. See Examples 1 through 3 52  a  23b
 1.3.1.3.15: Add or subtract as indicated. See Examples 1 through 3Subtract 14 f...
 1.3.1.3.16: Add or subtract as indicated. See Examples 1 through 3Subtract 9 fr...
 1.3.1.3.17: Multiply or divide as indicated. See Examples 4 and 5.5 # 12
 1.3.1.3.18: Multiply or divide as indicated. See Examples 4 and 5.3 # 8
 1.3.1.3.19: Multiply or divide as indicated. See Examples 4 and 5.17 # 0
 1.3.1.3.20: Multiply or divide as indicated. See Examples 4 and 5.5 # 0
 1.3.1.3.21: Multiply or divide as indicated. See Examples 4 and 5.02
 1.3.1.3.22: Multiply or divide as indicated. See Examples 4 and 5.20
 1.3.1.3.23: Multiply or divide as indicated. See Examples 4 and 5.93
 1.3.1.3.24: Multiply or divide as indicated. See Examples 4 and 5.205
 1.3.1.3.25: Multiply or divide as indicated. See Examples 4 and 5.124
 1.3.1.3.26: Multiply or divide as indicated. See Examples 4 and 5.366
 1.3.1.3.27: Multiply or divide as indicated. See Examples 4 and 5.3a  118 b
 1.3.1.3.28: Multiply or divide as indicated. See Examples 4 and 5.5a  150 b
 1.3.1.3.29: Multiply or divide as indicated. See Examples 4 and 5.1 0.721 0.82
 1.3.1.3.30: Multiply or divide as indicated. See Examples 4 and 5.1 0.921 0.52
 1.3.1.3.31: Multiply or divide as indicated. See Examples 4 and 5.9.1 , 1 1.32
 1.3.1.3.32: Multiply or divide as indicated. See Examples 4 and 5.22.5 , 1 2.52
 1.3.1.3.33: Multiply or divide as indicated. See Examples 4 and 5.41 221 12
 1.3.1.3.34: Multiply or divide as indicated. See Examples 4 and 5.51 321 22
 1.3.1.3.35: Evaluate each expression. See Example 6.72
 1.3.1.3.36: Evaluate each expression. See Example 6.1 722
 1.3.1.3.37: Evaluate each expression. See Example 6.1 622
 1.3.1.3.38: Evaluate each expression. See Example 6.62
 1.3.1.3.39: Evaluate each expression. See Example 6.1 223
 1.3.1.3.40: Evaluate each expression. See Example 6.23
 1.3.1.3.41: Evaluate each expression. See Example 6.a  13b3
 1.3.1.3.42: Evaluate each expression. See Example 6.a  12b4
 1.3.1.3.43: Find the following roots. See Examples 7 and 8.249
 1.3.1.3.44: Find the following roots. See Examples 7 and 8.281
 1.3.1.3.45: Find the following roots. See Examples 7 and 8. A49
 1.3.1.3.46: Find the following roots. See Examples 7 and 8. A425
 1.3.1.3.47: Find the following roots. See Examples 7 and 8.23 64
 1.3.1.3.48: Find the following roots. See Examples 7 and 8.25 32
 1.3.1.3.49: Find the following roots. See Examples 7 and 8.24 81
 1.3.1.3.50: Find the following roots. See Examples 7 and 8.23 1
 1.3.1.3.51: Find the following roots. See Examples 7 and 8.2100
 1.3.1.3.52: Find the following roots. See Examples 7 and 8.225
 1.3.1.3.53: Simplify each expression. See Examples 1 through 11.315  724
 1.3.1.3.54: Simplify each expression. See Examples 1 through 11.713  822
 1.3.1.3.55: Simplify each expression. See Examples 1 through 11.32 + 23
 1.3.1.3.56: Simplify each expression. See Examples 1 through 11.52  24
 1.3.1.3.57: Simplify each expression. See Examples 1 through 11.3.1  1 1.420.5
 1.3.1.3.58: Simplify each expression. See Examples 1 through 11.4.2  1 8.220.4
 1.3.1.3.59: Simplify each expression. See Examples 1 through 11.1 322 + 23
 1.3.1.3.60: Simplify each expression. See Examples 1 through 11.1 1522  24
 1.3.1.3.61: Simplify each expression. See Examples 1 through 11.8 , 4 # 2
 1.3.1.3.62: Simplify each expression. See Examples 1 through 11.20 , 5 # 4
 1.3.1.3.63: Simplify each expression. See Examples 1 through 11.8a 34b  8
 1.3.1.3.64: Simplify each expression. See Examples 1 through 11. 10a 25b  10
 1.3.1.3.65: Simplify each expression. See Examples 1 through 11.2  317  62 + ...
 1.3.1.3.66: Simplify each expression. See Examples 1 through 11.8  314  72 + ...
 1.3.1.3.67: Simplify each expression. See Examples 1 through 11.1 9 + 621 122...
 1.3.1.3.68: Simplify each expression. See Examples 1 through 11.1 1  221 322...
 1.3.1.3.69: Simplify each expression. See Examples 1 through 11.123 82142  12...
 1.3.1.3.70: Simplify each expression. See Examples 1 through 11.123 272152  1...
 1.3.1.3.71: Simplify each expression. See Examples 1 through 11.25  313  52 +...
 1.3.1.3.72: Simplify each expression. See Examples 1 through 11.10  314  522 ...
 1.3.1.3.73: Simplify each expression. See Examples 1 through 11.13  292  15 ...
 1.3.1.3.74: Simplify each expression. See Examples 1 through 11. 216  16  2....
 1.3.1.3.75: Simplify each expression. See Examples 1 through 11.0 3  9 0  0 ...
 1.3.1.3.76: Simplify each expression. See Examples 1 through 11.14 0  0 2  7...
 1.3.1.3.77: Simplify each expression. See Examples 1 through 11.312 + 125  7...
 1.3.1.3.78: Simplify each expression. See Examples 1 through 11.1  22132 + 1...
 1.3.1.3.79: Simplify each expression. See Examples 1 through 11.13# 9  73 +12# 4
 1.3.1.3.80: Simplify each expression. See Examples 1 through 11.15# 20  610 +1...
 1.3.1.3.81: Simplify each expression. See Examples 1 through 11.352 + 531  21...
 1.3.1.3.82: Simplify each expression. See Examples 1 through 11.251 + 337  41...
 1.3.1.3.83: Simplify each expression. See Examples 1 through 11.4280 + 1 + 1 ...
 1.3.1.3.84: Simplify each expression. See Examples 1 through 11.1 224 + 32120 ...
 1.3.1.3.85: Evaluate each expression when x = 9 and y = 2. See Example 12.9x  6y
 1.3.1.3.86: Evaluate each expression when x = 9 and y = 2. See Example 12.4x ...
 1.3.1.3.87: Evaluate each expression when x = 9 and y = 2. See Example 12.3y2
 1.3.1.3.88: Evaluate each expression when x = 9 and y = 2. See Example 12.7y2
 1.3.1.3.89: Evaluate each expression when x = 9 and y = 2. See Example 12.1xy ...
 1.3.1.3.90: Evaluate each expression when x = 9 and y = 2. See Example 12.y2x ...
 1.3.1.3.91: Evaluate each expression when x = 9 and y = 2. See Example 12.3 + ...
 1.3.1.3.92: Evaluate each expression when x = 9 and y = 2. See Example 12.5 + ...
 1.3.1.3.93: Evaluate each expression when x = 9 and y = 2. See Example 12.y3 +...
 1.3.1.3.94: Evaluate each expression when x = 9 and y = 2. See Example 12.y2 +...
 1.3.1.3.95: The algebraic expression 8 + 2y represents the perimeter ofa rectan...
 1.3.1.3.96: The algebraic expression pr2 represents the area of a circle with r...
 1.3.1.3.97: The algebraic expression 100x + 5000x represents the costper booksh...
 1.3.1.3.98: If c is degrees Celsius, the algebraic expression 1.8c + 32represen...
 1.3.1.3.99: Choose the fraction(s) equivalent to the given fraction. (There may...
 1.3.1.3.100: Choose the fraction(s) equivalent to the given fraction. (There may...
 1.3.1.3.101: Choose the fraction(s) equivalent to the given fraction. (There may...
 1.3.1.3.102: Choose the fraction(s) equivalent to the given fraction. (There may...
 1.3.1.3.103: Choose the fraction(s) equivalent to the given fraction. (There may...
 1.3.1.3.104: Choose the fraction(s) equivalent to the given fraction. (There may...
 1.3.1.3.105: Find the value of the expression when x1 = 2, x2 = 4, y1 = 3, y2 =...
 1.3.1.3.106: Find the value of the expression when x1 = 2, x2 = 4, y1 = 3, y2 =...
 1.3.1.3.107: Each circle below represents a whole, or 1. Determine the unknown f...
 1.3.1.3.108: Each circle below represents a whole, or 1. Determine the unknown f...
 1.3.1.3.109: Most of Mauna Kea, a volcano on Hawaii, lies below sea level. If th...
 1.3.1.3.110: The highest point on land on Earth is the top of Mt. Everest in the...
 1.3.1.3.111: Insert parentheses so that each expression simplifies to the given ...
 1.3.1.3.112: Insert parentheses so that each expression simplifies to the given ...
 1.3.1.3.113: The following graph is called a brokenline graph, or simply a line...
 1.3.1.3.114: The following graph is called a brokenline graph, or simply a line...
 1.3.1.3.115: The following graph is called a brokenline graph, or simply a line...
 1.3.1.3.116: The following graph is called a brokenline graph, or simply a line...
 1.3.1.3.117: The following graph is called a brokenline graph, or simply a line...
 1.3.1.3.118: The following graph is called a brokenline graph, or simply a line...
 1.3.1.3.119: Explain why 32 and 1 322 simplify to different numbers.
 1.3.1.3.120: Explain why 33 and 1 323 simplify to the same number.
 1.3.1.3.121: Use a calculator to approximate each square root. For Exercises 125...
 1.3.1.3.122: Use a calculator to approximate each square root. For Exercises 125...
 1.3.1.3.123: Use a calculator to approximate each square root. For Exercises 125...
 1.3.1.3.124: Use a calculator to approximate each square root. For Exercises 125...
 1.3.1.3.125: Use a calculator to approximate each square root. For Exercises 125...
 1.3.1.3.126: Use a calculator to approximate each square root. For Exercises 125...
Solutions for Chapter 1.3: Operations on Real Numbers and Order of Operations
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 1.3: Operations on Real Numbers and Order of Operations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 126 problems in chapter 1.3: Operations on Real Numbers and Order of Operations have been answered, more than 66212 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. Chapter 1.3: Operations on Real Numbers and Order of Operations includes 126 full stepbystep solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Outer product uv T
= column times row = rank one matrix.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).