 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
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 7.5.41: In Exercises 3964, rationalize each denominator.A11x
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 7.5.43: In Exercises 3964, rationalize each denominator.923y
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 7.5.45: In Exercises 3964, rationalize each denominator.123 2
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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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 7.5.113: 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...
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 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...
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 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 22738 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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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].

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!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Iterative method.
A sequence of steps intended to approach the desired solution.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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

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

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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