 7.5.1: Rationalize each denominator. See Examples 1 through 3.2227
 7.5.2: Rationalize each denominator. See Examples 1 through 3.2322
 7.5.3: Rationalize each denominator. See Examples 1 through 3.A15
 7.5.4: Rationalize each denominator. See Examples 1 through 3.A12
 7.5.5: Rationalize each denominator. See Examples 1 through 3.A4x
 7.5.6: Rationalize each denominator. See Examples 1 through 3.A25y
 7.5.7: Rationalize each denominator. See Examples 1 through 3.423 3
 7.5.8: Rationalize each denominator. See Examples 1 through 3.623 9
 7.5.9: Rationalize each denominator. See Examples 1 through 3.328x
 7.5.10: Rationalize each denominator. See Examples 1 through 3.5227a
 7.5.11: Rationalize each denominator. See Examples 1 through 3.323 4x2
 7.5.12: Rationalize each denominator. See Examples 1 through 3.523 3y
 7.5.13: Rationalize each denominator. See Examples 1 through 3.923a
 7.5.14: Rationalize each denominator. See Examples 1 through 3.x25
 7.5.15: Rationalize each denominator. See Examples 1 through 3.323 2
 7.5.16: Rationalize each denominator. See Examples 1 through 3.523 9
 7.5.17: Rationalize each denominator. See Examples 1 through 3.22327
 7.5.18: Rationalize each denominator. See Examples 1 through 3.522211
 7.5.19: Rationalize each denominator. See Examples 1 through 3.A2x5y
 7.5.20: Rationalize each denominator. See Examples 1 through 3.A13a2b
 7.5.21: Rationalize each denominator. See Examples 1 through 3.A3 35
 7.5.22: Rationalize each denominator. See Examples 1 through 3.A3 710
 7.5.23: Rationalize each denominator. See Examples 1 through 3.A3x50
 7.5.24: Rationalize each denominator. See Examples 1 through 3.B11y45
 7.5.25: Rationalize each denominator. See Examples 1 through 3.1212z
 7.5.26: Rationalize each denominator. See Examples 1 through 3.1232x
 7.5.27: Rationalize each denominator. See Examples 1 through 3.23 2y223 9x2
 7.5.28: Rationalize each denominator. See Examples 1 through 3.23 3x23 4y4
 7.5.29: Rationalize each denominator. See Examples 1 through 3.A4 818
 7.5.30: Rationalize each denominator. See Examples 1 through 3.A4 19
 7.5.31: Rationalize each denominator. See Examples 1 through 3.A4 169x7
 7.5.32: Rationalize each denominator. See Examples 1 through 3.A5 32m6n13
 7.5.33: Rationalize each denominator. See Examples 1 through 3.5a25 8a9b11
 7.5.34: Rationalize each denominator. See Examples 1 through 3.9y24 4y9
 7.5.35: Write the conjugate of each expression22 + x
 7.5.36: Write the conjugate of each expression23 + y
 7.5.37: Write the conjugate of each expression5  2a
 7.5.38: Write the conjugate of each expression6  2b
 7.5.39: Write the conjugate of each expression725 + 82x
 7.5.40: Write the conjugate of each expression922  62y
 7.5.41: Rationalize each denominator. See Example 4.62  27
 7.5.42: Rationalize each denominator. See Example 4.327  4
 7.5.43: Rationalize each denominator. See Example 4.72x  3
 7.5.44: Rationalize each denominator. See Example 4.82y + 4
 7.5.45: Rationalize each denominator. See Example 4.22  2322 + 23
 7.5.46: Rationalize each denominator. See Example 4.23 + 2422  23
 7.5.47: Rationalize each denominator. See Example 4.2a + 122a  2b
 7.5.48: Rationalize each denominator. See Example 4.22a  322a + 2b
 7.5.49: Rationalize each denominator. See Example 4.81 + 210
 7.5.50: Rationalize each denominator. See Example 4.326  2
 7.5.51: Rationalize each denominator. See Example 4.2x2x + 2y
 7.5.52: Rationalize each denominator. See Example 4.22a22x  2y
 7.5.53: Rationalize each denominator. See Example 4.223 + 26423  26
 7.5.54: Rationalize each denominator. See Example 4.425 + 22225  22
 7.5.55: Rationalize each numerator. See Examples 5 and 6.A53
 7.5.56: Rationalize each numerator. See Examples 5 and 6.A32
 7.5.57: Rationalize each numerator. See Examples 5 and 6.A185
 7.5.58: Rationalize each numerator. See Examples 5 and 6.A127
 7.5.59: Rationalize each numerator. See Examples 5 and 6.24x7
 7.5.60: Rationalize each numerator. See Examples 5 and 6.23x56
 7.5.61: Rationalize each numerator. See Examples 5 and 6.23 5y223 4x
 7.5.62: Rationalize each numerator. See Examples 5 and 6.23 4x23 z4
 7.5.63: Rationalize each numerator. See Examples 5 and 6.A25
 7.5.64: Rationalize each numerator. See Examples 5 and 6.A37
 7.5.65: Rationalize each numerator. See Examples 5 and 6.22x11
 7.5.66: Rationalize each numerator. See Examples 5 and 6.2y7
 7.5.67: Rationalize each numerator. See Examples 5 and 6.A3 78
 7.5.68: Rationalize each numerator. See Examples 5 and 6.A3 252
 7.5.69: Rationalize each numerator. See Examples 5 and 6.23 3x510
 7.5.70: Rationalize each numerator. See Examples 5 and 6.3 9y7
 7.5.71: Rationalize each numerator. See Examples 5 and 6.B18x4y63z
 7.5.72: Rationalize each numerator. See Examples 5 and 6.B8x5y2z
 7.5.73: Rationalize each numerator. See Example 7. 2  2116
 7.5.74: Rationalize each numerator. See Example 7.215 + 12
 7.5.75: Rationalize each numerator. See Example 7.2  275
 7.5.76: Rationalize each numerator. See Example 7.25 + 222
 7.5.77: Rationalize each numerator. See Example 7.2x + 32x
 7.5.78: Rationalize each numerator. See Example 7. 5 + 2222x
 7.5.79: Rationalize each numerator. See Example 7.22  122 + 1
 7.5.80: Rationalize each numerator. See Example 7.28  2322 + 23
 7.5.81: Rationalize each numerator. See Example 7.2x + 12x  1
 7.5.82: Rationalize each numerator. See Example 7.2x + 2y2x  2y
 7.5.83: Solve each equation. See Sections 2.1 and 5.8.2x  7 = 31x  42
 7.5.84: Solve each equation. See Sections 2.1 and 5.8. 9x  4 = 71x  22
 7.5.85: Solve each equation. See Sections 2.1 and 5.8.1x  6212x + 12 = 0
 7.5.86: Solve each equation. See Sections 2.1 and 5.8.1y + 2215y + 42 = 0
 7.5.87: Solve each equation. See Sections 2.1 and 5.8.x2  8x = 12
 7.5.88: Solve each equation. See Sections 2.1 and 5.8.x3 = x
 7.5.89: The formula of the radius r of a sphere with surface area A isr = A...
 7.5.90: The formula for the radius r of a cone with height 7 centimetersand...
 7.5.91: Given25y3212x3, rationalize the denominator by following parts (a) ...
 7.5.92: Given23 5y23 4, rationalize the denominator by following parts (a) ...
 7.5.93: Determine the smallest number both the numerator and denominator sh...
 7.5.94: Determine the smallest number both the numerator and denominator sh...
 7.5.95: When rationalizing the denominator of 2527, explain why both the nu...
 7.5.96: When rationalizing the numerator of 2527, explain why both the nume...
 7.5.97: Explain why rationalizing the denominator does not change the value...
 7.5.98: Explain why rationalizing the numerator does not change the value o...
Solutions for Chapter 7.5: Rationalizing Denominators and Numerators of Radical Expressions
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 7.5: Rationalizing Denominators and Numerators of Radical Expressions
Get Full SolutionsChapter 7.5: Rationalizing Denominators and Numerators of Radical Expressions includes 98 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Since 98 problems in chapter 7.5: Rationalizing Denominators and Numerators of Radical Expressions have been answered, more than 85502 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046.

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.

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

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

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.

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.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

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