 7.4.1: 523 + 823 = ( + )23 =
 7.4.2: 227  212 = 2 # 3  2 # 3 = 23  23 =
 7.4.3: 23 54 + 23 16 = 23 # 2 + 23 # 2 = 23 2 + 23 2 =
 7.4.4: If 2 n a and 2 n b are real numbers and b 0, the quotient rule for ...
 7.4.5: 3 8 27 = 23 23 =
 7.4.6: . If x 7 0, 272x3 22x = B = 2 = .
 7.4.7: n Exercises 110, add or subtract as indicated.3213  225  2213 + 425
 7.4.8: n Exercises 110, add or subtract as indicated.8217  5219  6217 + ...
 7.4.9: n Exercises 110, add or subtract as indicated.325  23 x + 425 + 323 x
 7.4.10: n Exercises 110, add or subtract as indicated.627  23 x + 227 + 523 x
 7.4.11: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.12: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.13: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.14: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.15: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.16: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.17: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.18: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.19: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.20: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.21: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.22: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.23: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.24: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.25: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.26: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.27: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.28: In Exercises 1128, add or subtract as indicated. You will need to s...
 7.4.29: In Exercises 2944, simplify using the quotient rule.A114
 7.4.30: In Exercises 2944, simplify using the quotient rule.A1925
 7.4.31: In Exercises 2944, simplify using the quotient rule.A3 1927
 7.4.32: In Exercises 2944, simplify using the quotient rule.A3 1164
 7.4.33: In Exercises 2944, simplify using the quotient rule.Bx236y8
 7.4.34: In Exercises 2944, simplify using the quotient rule.Bx2144y12
 7.4.35: In Exercises 2944, simplify using the quotient rule.B8x325y6
 7.4.36: In Exercises 2944, simplify using the quotient rule.B50x381y8
 7.4.37: In Exercises 2944, simplify using the quotient rule.B3 x48y3
 7.4.38: In Exercises 2944, simplify using the quotient rule.B3 x5125y3
 7.4.39: In Exercises 2944, simplify using the quotient rule.B3 50x827y12
 7.4.40: In Exercises 2944, simplify using the quotient rule.B3 81x88y15
 7.4.41: In Exercises 2944, simplify using the quotient rule.B4 9y6x8
 7.4.42: In Exercises 2944, simplify using the quotient rule.B4 13y7x12
 7.4.43: In Exercises 2944, simplify using the quotient rule.B5 64x13y20
 7.4.44: In Exercises 2944, simplify using the quotient rule.B5 64x14y15
 7.4.45: In Exercises 4566, divide and, if possible, simplify.24025
 7.4.46: In Exercises 4566, divide and, if possible, simplify.2200210
 7.4.47: In Exercises 4566, divide and, if possible, simplify.23 4823 6
 7.4.48: In Exercises 4566, divide and, if possible, simplify.23 5423 2
 7.4.49: In Exercises 4566, divide and, if possible, simplify.254x326x
 7.4.50: In Exercises 4566, divide and, if possible, simplify.272x322x
 7.4.51: In Exercises 4566, divide and, if possible, simplify.2x5y31xy
 7.4.52: In Exercises 4566, divide and, if possible, simplify.2x7y62x3y2
 7.4.53: In Exercises 4566, divide and, if possible, simplify.2200x3210x1
 7.4.54: In Exercises 4566, divide and, if possible, simplify.2500x3210x1
 7.4.55: In Exercises 4566, divide and, if possible, simplify.248a8b723a2b3
 7.4.56: In Exercises 4566, divide and, if possible, simplify.254a7b1123a4b2
 7.4.57: In Exercises 4566, divide and, if possible, simplify.272xy222
 7.4.58: In Exercises 4566, divide and, if possible, simplify.250xy222
 7.4.59: In Exercises 4566, divide and, if possible, simplify.3 24x3y523 3y2
 7.4.60: In Exercises 4566, divide and, if possible, simplify.23 250x5y323 2x3
 7.4.61: In Exercises 4566, divide and, if possible, simplify.24 32x10y824 2...
 7.4.62: In Exercises 4566, divide and, if possible, simplify.25 96x12y1125 ...
 7.4.63: In Exercises 4566, divide and, if possible, simplify.23 x2 + 5x + 6...
 7.4.64: In Exercises 4566, divide and, if possible, simplify.23 x2 + 7x + 1...
 7.4.65: In Exercises 4566, divide and, if possible, simplify.23 a3 + b323 a...
 7.4.66: In Exercises 4566, divide and, if possible, simplify.23 a3  b323 a...
 7.4.67: In Exercises 6776, perform the indicated operations.2325+2187
 7.4.68: In Exercises 6776, perform the indicated operations.2272+2757
 7.4.69: In Exercises 6776, perform the indicated operations.3x28xy2  5y232...
 7.4.70: In Exercises 6776, perform the indicated operations.6x23xy2  4x222...
 7.4.71: In Exercises 6776, perform the indicated operations.522x3 +30x3224x...
 7.4.72: In Exercises 6776, perform the indicated operations.722x3 +40x32150...
 7.4.73: In Exercises 6776, perform the indicated operations.2x275xy  281xy...
 7.4.74: In Exercises 6776, perform the indicated operations.528x2y3  9x226...
 7.4.75: In Exercises 6776, perform the indicated operations.15x423 80x3y25x...
 7.4.76: In Exercises 6776, perform the indicated operations.16x423 48x3y28x...
 7.4.77: In Exercises 7780, find 1 f g 2(x) and the domain of f g. Express e...
 7.4.78: In Exercises 7780, find 1 f g 2(x) and the domain of f g. Express e...
 7.4.79: In Exercises 7780, find 1 f g 2(x) and the domain of f g. Express e...
 7.4.80: In Exercises 7780, find 1 f g 2(x) and the domain of f g. Express e...
 7.4.81: Exercises 8184 involve the perimeter and area of various geometric ...
 7.4.82: Exercises 8184 involve the perimeter and area of various geometric ...
 7.4.83: Exercises 8184 involve the perimeter and area of various geometric ...
 7.4.84: Exercises 8184 involve the perimeter and area of various geometric ...
 7.4.85: America is getting older. The graph shows the projected elderly U.S...
 7.4.86: What does travel in space have to do with radicals? Imagine that in...
 7.4.87: What are like radicals? Give an example with your explanation.
 7.4.88: Explain how to add like radicals. Give an example with your explana...
 7.4.89: If only like radicals can be combined, why is it possible to add 22...
 7.4.90: Explain how to simplify a radical expression using the quotient rul...
 7.4.91: Explain how to divide radical expressions with the same index.
 7.4.92: In Exercise 85, use the data displayed by the bar graph to explain ...
 7.4.93: Use a calculator to provide numerical support for any four exercise...
 7.4.94: In Exercises 9496, determine if each operation is performed correct...
 7.4.95: In Exercises 9496, determine if each operation is performed correct...
 7.4.96: In Exercises 9496, determine if each operation is performed correct...
 7.4.97: Make Sense? In Exercises 97100, determine whether each statement ma...
 7.4.98: Make Sense? In Exercises 97100, determine whether each statement ma...
 7.4.99: Make Sense? In Exercises 97100, determine whether each statement ma...
 7.4.100: Make Sense? In Exercises 97100, determine whether each statement ma...
 7.4.101: In Exercises 101104, determine whether each statement is true or fa...
 7.4.102: In Exercises 101104, determine whether each statement is true or fa...
 7.4.103: In Exercises 101104, determine whether each statement is true or fa...
 7.4.104: In Exercises 101104, determine whether each statement is true or fa...
 7.4.105: If an irrational number is decreased by 2218  250, the result is 2...
 7.4.106: Simplify: 220 3 + 245 4  280.
 7.4.107: Simplify: 6249xy 2ab2 7236x3y52a9b1 .
 7.4.108: Solve: 2(3x  1)  4 = 2x  (6  x). (Section 1.4, Example 3)
 7.4.109: Factor: x2  8xy + 12y2 . (Section 5.4, Example 4)
 7.4.110: Add: 2 x2 + 5x + 6 + 3x x2 + 6x + 9 . (Section 6.2, Example 6)
 7.4.111: Exercises 111113 will help you prepare for the material covered in ...
 7.4.112: Exercises 111113 will help you prepare for the material covered in ...
 7.4.113: Exercises 111113 will help you prepare for the material covered in ...
Solutions for Chapter 7.4: Adding, Subtracting, and Dividing Radical Expressions
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 7.4: Adding, Subtracting, and Dividing Radical Expressions
Get Full SolutionsThis textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.4: Adding, Subtracting, and Dividing Radical Expressions includes 113 full stepbystep solutions. Since 113 problems in chapter 7.4: Adding, Subtracting, and Dividing Radical Expressions have been answered, more than 75164 students have viewed full stepbystep solutions from this chapter.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

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.

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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

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 N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.