 6.2.1: Rewrite each expression with exponents. a. (7)(7)(7)(7)(7)(7)(7)(7)...
 6.2.2: A bacteria culture grows at a rate of 20% each day. There are 450 b...
 6.2.3: Match each equation with a table of values. a. y = 4(2)x b. y = 4(0...
 6.2.4: Match each recursive routine with the equation that gives the same ...
 6.2.5: For each table, find the value of the constants a and b such that y...
 6.2.6: The equation y = 500(1 + 0.04)x models the amount of money in a sav...
 6.2.7: Run the calculator program INOUTEXP and play the easylevel game fi...
 6.2.8: APPLICATION A credit card account is essentially a loan. A constant...
 6.2.9: APPLICATION Phil purchases a used truck for $11,500. The value of t...
 6.2.10: Draw a starting line segment 2 cm long on a sheet of paper. a.Draw ...
 6.2.11: Run the calculator program INOUTEXP and play the medium or difficu...
 6.2.12: Fold a sheet of paper in half. You should have two layers. Fold it ...
 6.2.13: APPLICATION Phils friend Shawna buys an antique car for $5,000. She...
 6.2.14: Invent a situation that could be modeled by each equation below. Sk...
 6.2.15: Consider the recursive routine {0, 100} {Ans(1) + 1, Ans(2) (1 0.03...
 6.2.16: . Look at this step pattern. In the first figure, which has one ste...
Solutions for Chapter 6.2: Exponential Equations
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 6.2: Exponential Equations
Get Full SolutionsChapter 6.2: Exponential Equations includes 16 full stepbystep solutions. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Since 16 problems in chapter 6.2: Exponential Equations have been answered, more than 8873 students have viewed full stepbystep solutions from this chapter. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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).

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

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).

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

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.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = 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).

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.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.