 P.2.1: In the exponential form an, n is the _______ and a is the _______ .
 P.2.2: One of the two equal factors of a number is called a _______ of the...
 P.2.3: The _______ of a number a is the nth root that has the same sign as a
 P.2.4: In the radical form n a, the positive integer n is called the _____...
 P.2.5: Creating a radicalfree denominator is known as _______ the denomin...
 P.2.6: In the expression bmn, m denotes the _______ to which the base is r...
 P.2.7: What is the conjugate of 2 + 35?
 P.2.8: When is an expression involving radicals in simplest form?
 P.2.9: . Is 10.678 103 written in scientific notation?
 P.2.10: Is 64 a perfect square, a perfect cube, or both?
 P.2.11: In Exercises 1118, evaluate each expression. (a) 3 33
 P.2.12: In Exercises 1118, evaluate each expression. a) 53 52 (b) 42 4
 P.2.13: In Exercises 1118, evaluate each expression. (a) (43)0 (b) 6 63
 P.2.14: In Exercises 1118, evaluate each expression. (a) 24(2)5 (b) 70
 P.2.15: In Exercises 1118, evaluate each expression. (a) 24(2)5 (b) 70
 P.2.16: In Exercises 1118, evaluate each expression. . (a) (23 32)2 (b) (58)2
 P.2.17: In Exercises 1118, evaluate each expression. (a) 21 + 31 (b) (31)3
 P.2.18: In Exercises 1118, evaluate each expression. a) 4 32 22 31 (b) 31 + 22
 P.2.19: In Exercises 1924, evaluate the expression for the value of x. 2x3,...
 P.2.20: In Exercises 1924, evaluate the expression for the value of x. 3x4,...
 P.2.21: In Exercises 1924, evaluate the expression for the value of x. 5(x)...
 P.2.22: In Exercises 1924, evaluate the expression for the value of x. 6x0 ...
 P.2.23: In Exercises 1924, evaluate the expression for the value of x. 7x2,...
 P.2.24: In Exercises 1924, evaluate the expression for the value of x. 20x2...
 P.2.25: In Exercises 2532, simplify each expression. . (a) x3(x2) (b) x4(3x5
 P.2.26: In Exercises 2532, simplify each expression. (a) 4z4(2z3) (b) 5x3(4x2)
 P.2.27: In Exercises 2532, simplify each expression. (a) (3x)2 (b) (4x3)0, x 0
 P.2.28: In Exercises 2532, simplify each expression. (a) 6z2(2z5)4 (b) (3x5...
 P.2.29: In Exercises 2532, simplify each expression. (a) 7x2 x3 (b) 12(x + ...
 P.2.30: In Exercises 2532, simplify each expression. a) r5 r9 (b) 3x2y4 15(...
 P.2.31: In Exercises 2532, simplify each expression. (a) [(x2y2)1]1 (b) (a2...
 P.2.32: In Exercises 2532, simplify each expression. a) (4y)3(3y)4(b) (5x2z...
 P.2.33: In Exercises 3338, use a calculator to evaluate the expression. (Ro...
 P.2.34: In Exercises 3338, use a calculator to evaluate the expression. (Ro...
 P.2.35: In Exercises 3338, use a calculator to evaluate the expression. (Ro...
 P.2.36: In Exercises 3338, use a calculator to evaluate the expression. (Ro...
 P.2.37: In Exercises 3338, use a calculator to evaluate the expression. (Ro...
 P.2.38: In Exercises 3338, use a calculator to evaluate the expression. (Ro...
 P.2.39: In Exercises 3952, write the number in scientific notation. 973.50
 P.2.40: In Exercises 3952, write the number in scientific notation. 28,022.2
 P.2.41: In Exercises 3952, write the number in scientific notation. 10,252.484
 P.2.42: In Exercises 3952, write the number in scientific notation. 525,252...
 P.2.43: In Exercises 3952, write the number in scientific notation. . 1110.25
 P.2.44: In Exercises 3952, write the number in scientific notation. 5,222,145
 P.2.45: In Exercises 3952, write the number in scientific notation. 0.0002485
 P.2.46: In Exercises 3952, write the number in scientific notation. 0.0000025
 P.2.47: In Exercises 3952, write the number in scientific notation. 0.0000025
 P.2.48: In Exercises 3952, write the number in scientific notation. 0.00012...
 P.2.49: In Exercises 3952, write the number in scientific notation. Land ar...
 P.2.50: In Exercises 3952, write the number in scientific notation. Light y...
 P.2.51: In Exercises 3952, write the number in scientific notation. Relativ...
 P.2.52: In Exercises 3952, write the number in scientific notation. One mic...
 P.2.53: In Exercises 5360, write the number in decimal notation. 1.08 104
 P.2.54: In Exercises 5360, write the number in decimal notation. 4.816 108
 P.2.55: In Exercises 5360, write the number in decimal notation. 7.65 107
 P.2.56: In Exercises 5360, write the number in decimal notation. 5.098 1010
 P.2.57: In Exercises 5360, write the number in decimal notation. Mean SAT M...
 P.2.58: In Exercises 5360, write the number in decimal notation. Core tempe...
 P.2.59: In Exercises 5360, write the number in decimal notation. Width of a...
 P.2.60: In Exercises 5360, write the number in decimal notation. Charge of ...
 P.2.61: In Exercises 6166, evaluate the expression without using a calculator.
 P.2.62: In Exercises 6166, evaluate the expression without using a calculator.
 P.2.63: In Exercises 6166, evaluate the expression without using a calculator.
 P.2.64: In Exercises 6166, evaluate the expression without using a calculator.
 P.2.65: In Exercises 6166, evaluate the expression without using a calculator.
 P.2.66: In Exercises 6166, evaluate the expression without using a calculator.
 P.2.67: In Exercises 6770, use a calculator to evaluate each expression. (R...
 P.2.68: In Exercises 6770, use a calculator to evaluate each expression. (R...
 P.2.69: In Exercises 6770, use a calculator to evaluate each expression. (R...
 P.2.70: In Exercises 6770, use a calculator to evaluate each expression. (R...
 P.2.71: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.72: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.73: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.74: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.75: In Exerises 7178, evaluate the expression (if possible) without usi...
 P.2.76: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.77: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.78: In Exercises 7178, evaluate the expression (if possible) without us...
 P.2.79: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.80: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.81: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.82: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.83: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.84: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.85: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.86: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.87: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.88: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.89: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.90: In Exercises 7990, use a calculator to approximate the value of the...
 P.2.91: In Exercises 9194, use the properties of radicals to simplify each ...
 P.2.92: In Exercises 9194, use the properties of radicals to simplify each ...
 P.2.93: In Exercises 9194, use the properties of radicals to simplify each ...
 P.2.94: In Exercises 9194, use the properties of radicals to simplify each ...
 P.2.95: In Exercises 95102, simplify each expression. . (a) 45 (b) 3 32a2b2
 P.2.96: In Exercises 95102, simplify each expression. (a) 3 54 (b) 32x3y4
 P.2.97: In Exercises 95102, simplify each expression. (a) 3 16x5 (b) 75x2y4
 P.2.98: In Exercises 95102, simplify each expression. (a) 4 3x4y2 (b) 5 160...
 P.2.99: In Exercises 95102, simplify each expression. (a) 250 + 128 (b) 103...
 P.2.100: In Exercises 95102, simplify each expression. . (a) 5x 3x (b) 29y + 0y
 P.2.101: In Exercises 95102, simplify each expression. (a) 3x + 1 + 10x + 1 ...
 P.2.102: In Exercises 95102, simplify each expression. (a) 510x2 90x2 (b) 8 ...
 P.2.103: In Exercises 103106, complete the statement with <, =, or > 3 11311
 P.2.104: In Exercises 103106, complete the statement with <, =, or > 5 + 35 + 3
 P.2.105: In Exercises 103106, complete the statement with <, =, or > . 532 + 22
 P.2.106: In Exercises 103106, complete the statement with <, =, or > 532 + 4
 P.2.107: In Exercises 107110, rationalize the denominator of the expression....
 P.2.108: In Exercises 107110, rationalize the denominator of the expression....
 P.2.109: In Exercises 107110, rationalize the denominator of the expression....
 P.2.110: In Exercises 107110, rationalize the denominator of the expression....
 P.2.111: In Exercises 111114, rationalize the numerator of the expression. T...
 P.2.112: In Exercises 111114, rationalize the numerator of the expression. T...
 P.2.113: In Exercises 111114, rationalize the numerator of the expression. T...
 P.2.114: In Exercises 111114, rationalize the numerator of the expression. T...
 P.2.115: In Exercises 115 and 116, reduce the index of each radical and rewr...
 P.2.116: In Exercises 115 and 116, reduce the index of each radical and rewr...
 P.2.117: In Exercises 117124, fill in the missing form of the expression. 364
 P.2.118: In Exercises 117124, fill in the missing form of the expression. 144
 P.2.119: In Exercises 117124, fill in the missing form of the expression. 1/32
 P.2.120: In Exercises 117124, fill in the missing form of the expression. 316
 P.2.121: In Exercises 117124, fill in the missing form of the expression. (243
 P.2.122: In Exercises 117124, fill in the missing form of the expression. 34...
 P.2.123: In Exercises 117124, fill in the missing form of the expression. 34...
 P.2.124: In Exercises 117124, fill in the missing form of the expression. 1654
 P.2.125: In Exercises 125132, simplify the expression. 2x2)32 212x4
 P.2.126: In Exercises 125132, simplify the expression. x43y23 (xy)13
 P.2.127: In Exercises 125132, simplify the expression. x43y23 (xy)13
 P.2.128: In Exercises 125132, simplify the expression. 512 5x52 (5x)32
 P.2.129: In Exercises 125132, simplify the expression. 235
 P.2.130: In Exercises 125132, simplify the expression. 9 4) 1
 P.2.131: In Exercises 125132, simplify the expression. ( 1 27) 1
 P.2.132: In Exercises 125132, simplify the expression. ( 1 125) 4
 P.2.133: In Exercises 133136, write each expression as a single radical. The...
 P.2.134: In Exercises 133136, write each expression as a single radical. The...
 P.2.135: In Exercises 133136, write each expression as a single radical. The...
 P.2.136: In Exercises 133136, write each expression as a single radical. The...
 P.2.137: The table shows the 2013 estimated populations and gross domestic p...
 P.2.138: There were 2.51 108 tons of municipal waste generated in 2012. Find...
 P.2.139: A funnel is filled with hydrochloric acid to a height of h centimet...
 P.2.140: In Exercises 140143, determine whether the statement is true or fal...
 P.2.141: In Exercises 140143, determine whether the statement is true or fal...
 P.2.142: In Exercises 140143, determine whether the statement is true or fal...
 P.2.143: In Exercises 140143, determine whether the statement is true or fal...
 P.2.144: Package A is a cube with a volume of 500 cubic inches. Package B is...
 P.2.145: Verify that a0 = 1, a 0. (Hint: Use the property of exponents aman ...
 P.2.146: List all possible digits that occur in the units place of the squar...
 P.2.147: Square the real number 53 and note that the radical is eliminated f...
Solutions for Chapter P.2: Prerequisites
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter P.2: Prerequisites
Get Full SolutionsSince 147 problems in chapter P.2: Prerequisites have been answered, more than 26698 students have viewed full stepbystep solutions from this chapter. Chapter P.2: Prerequisites includes 147 full stepbystep solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by Patricia and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.