 7.5.1: In Exercises 138, multiply as indicated. If possible, simplify any ...
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 7.5.39: In Exercises 3964, rationalize each denominator.2225
 7.5.40: In Exercises 3964, rationalize each denominator.2723
 7.5.41: In Exercises 3964, rationalize each denominator.A11x
 7.5.42: In Exercises 3964, rationalize each denominator.A6x
 7.5.43: In Exercises 3964, rationalize each denominator.923y
 7.5.44: In Exercises 3964, rationalize each denominator.1223y
 7.5.45: In Exercises 3964, rationalize each denominator.123 2
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 7.5.49: In Exercises 3964, rationalize each denominator.A3 23
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 7.5.51: In Exercises 3964, rationalize each denominator.423 x
 7.5.52: In Exercises 3964, rationalize each denominator.723 x
 7.5.53: In Exercises 3964, rationalize each denominator.A3 2y2
 7.5.54: In Exercises 3964, rationalize each denominator.A3 5y2
 7.5.55: In Exercises 3964, rationalize each denominator.723 2x2
 7.5.56: In Exercises 3964, rationalize each denominator.1023 4x2
 7.5.57: In Exercises 3964, rationalize each denominator.A3 2xy2
 7.5.58: In Exercises 3964, rationalize each denominator.A3 3xy2
 7.5.59: In Exercises 3964, rationalize each denominator.324 x
 7.5.60: In Exercises 3964, rationalize each denominator.524 x
 7.5.61: In Exercises 3964, rationalize each denominator.625 8x3
 7.5.62: In Exercises 3964, rationalize each denominator.1025 16x2
 7.5.63: In Exercises 3964, rationalize each denominator.2x2y25 4x2y4
 7.5.64: In Exercises 3964, rationalize each denominator.3xy225 8xy3
 7.5.65: In Exercises 6574, simplify each radical expression and then ration...
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 7.5.105: In Exercises 105112, add or subtract as indicated. Begin by rationa...
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 7.5.112: In Exercises 105112, add or subtract as indicated. Begin by rationa...
 7.5.113: Let f(x) = x2  6x  4. Find f13  2132.
 7.5.114: Let f(x) = x2 + 4x  2. Find f 12 + 262.
 7.5.115: Let f(x) = 29 + x. Find f 13252 # f 13252.
 7.5.116: Let f(x) = x2 . Find f1 2a + 1  2a  12.
 7.5.117: The early Greeks believed that the most pleasing of all rectangles ...
 7.5.118: In the Peanuts cartoon shown in the section opener on page 542, Woo...
 7.5.119: In Exercises 119120, write expressions for the perimeter and area o...
 7.5.120: In Exercises 119120, write expressions for the perimeter and area o...
 7.5.121: The Pythagorean Theorem for right triangles tells us that the lengt...
 7.5.122: The Pythagorean Theorem for right triangles tells us that the lengt...
 7.5.123: Explain how to perform this multiplication: 221 27 + 2102.
 7.5.124: Explain how to perform this multiplication: 12 + 23214 + 232.
 7.5.125: Explain how to perform this multiplication: 12 + 2322 .
 7.5.126: What are conjugates? Give an example with your explanation.
 7.5.127: Describe how to multiply conjugates.
 7.5.128: Describe what it means to rationalize a denominator. Use both 1 25 ...
 7.5.129: When a radical expression has its denominator rationalized, we chan...
 7.5.130: Square the real number 2 23 . Observe that the radical is eliminate...
 7.5.131: In Exercises 131134, determine if each operation is performed corre...
 7.5.132: In Exercises 131134, determine if each operation is performed corre...
 7.5.133: In Exercises 131134, determine if each operation is performed corre...
 7.5.134: In Exercises 131134, determine if each operation is performed corre...
 7.5.135: Make Sense? In Exercises 135138, determine whether each statement m...
 7.5.136: Make Sense? In Exercises 135138, determine whether each statement m...
 7.5.137: Make Sense? In Exercises 135138, determine whether each statement m...
 7.5.138: Make Sense? In Exercises 135138, determine whether each statement m...
 7.5.139: In Exercises 139142, determine whether each statement is true or fa...
 7.5.140: In Exercises 139142, determine whether each statement is true or fa...
 7.5.141: In Exercises 139142, determine whether each statement is true or fa...
 7.5.142: In Exercises 139142, determine whether each statement is true or fa...
 7.5.143: Solve: 7[(2x  5)  (x + 1)] = 1 27 + 221 27  22.
 7.5.144: Simplify: 1 22 + 13 + 22  1322
 7.5.145: Rationalize the denominator: 1 22 + 23 + 24 .
 7.5.146: Add: 2 x  2 + 3 x2  4 . (Section 6.2, Example 6)
 7.5.147: Solve: 2 x  2 + 3 x2  4 = 0. (Section 6.6, Example 5)
 7.5.148: If f(x) = x4  3x2  2x + 5, use synthetic division and the Remaind...
 7.5.149: Exercises 149151 will help you prepare for the material covered in ...
 7.5.150: Exercises 149151 will help you prepare for the material covered in ...
 7.5.151: Exercises 149151 will help you prepare for the material covered in ...
Solutions for Chapter 7.5: Multiplying with More Than One Term and Rationalizing Denominators
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 7.5: Multiplying with More Than One Term and Rationalizing Denominators
Get Full SolutionsChapter 7.5: Multiplying with More Than One Term and Rationalizing Denominators includes 151 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 151 problems in chapter 7.5: Multiplying with More Than One Term and Rationalizing Denominators have been answered, more than 10452 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

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 .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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 ().

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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