 7.5.7.1.472: Fill in each blank so that the resulting statement is true. A ratio...
 7.5.7.1.473: Fill in each blank so that the resulting statement is true. 3 4 1 2...
 7.5.7.1.474: Fill in each blank so that the resulting statement is true. 5 x x 3...
 7.5.7.1.475: Fill in each blank so that the resulting statement is true. 2 x 3 x...
 7.5.7.1.476: In Exercises 140, simplify each complex rational expression by the ...
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 7.5.7.1.516: In Exercises 4148, simplify each complex rational expression. 6 x2 ...
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 7.5.7.1.524: The average rate on a roundtrip commute having a oneway distance d...
 7.5.7.1.525: If two electrical resistors with resistances R1 and R2 are connecte...
 7.5.7.1.526: The bar graph shows the amount, in billions of dollars, that the Un...
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 7.5.7.1.528: What is a complex rational expression? Give an example with your ex...
 7.5.7.1.529: Describe two ways to simplify 3 x 2 x2 1 x2 2 x .
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 7.5.7.1.531: In Exercises 5659, determine whether each statement makes sense or ...
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 7.5.7.1.539: In one short sentence, five words or less, explain what 1 x 1 x2 1 ...
 7.5.7.1.540: In Exercises 6566, simplify completely. 2y 2 2 y y 1 1 y
 7.5.7.1.541: In Exercises 6566, simplify completely. 1 1 y 6 y2 1 5 y 6 y2 1 1 y...
 7.5.7.1.542: In Exercises 6769, use the GRAPH or TABLE feature of a graphing uti...
 7.5.7.1.543: In Exercises 6769, use the GRAPH or TABLE feature of a graphing uti...
 7.5.7.1.544: In Exercises 6769, use the GRAPH or TABLE feature of a graphing uti...
 7.5.7.1.545: Factor completely: 2x3 20x2 50x. (Section 6.5, Example 2)
 7.5.7.1.546: Solve: 2 3(x 2) 5(x 5) 1. (Section 2.3, Example 3)
 7.5.7.1.547: Multiply: (x y)(x2 xy y2). (Section 5.2, Example 7)
 7.5.7.1.548: Exercises 7375 will help you prepare for the material covered in th...
 7.5.7.1.549: Exercises 7375 will help you prepare for the material covered in th...
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Solutions for Chapter 7.5: Complex Rational Expressions
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 7.5: Complex Rational Expressions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 79 problems in chapter 7.5: Complex Rational Expressions have been answered, more than 71372 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Chapter 7.5: Complex Rational Expressions includes 79 full stepbystep solutions. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941.

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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 ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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