 6.3.1: Use the properties of exponents to rewrite each expression. Use you...
 6.3.2: Write each expression in expanded form. Then rewrite the product in...
 6.3.3: Rewrite each expression with a single exponent. a. (35)8 b. (73)4 c...
 6.3.4: Use the properties of exponents to rewrite each expression. a. (rt)...
 6.3.5: An algebra class had this problem on a quiz: Find the value of 2x2 ...
 6.3.6: Match expressions from this list that are equivalent but written in...
 6.3.7: Evaluate each expression in Exercise 6 using an xvalue of 4.7.
 6.3.8: Use the properties of exponents to rewrite each expression. Use you...
 6.3.9: Cal and Als teacher asked them, What do you get when you square neg...
 6.3.10: Evaluate 2x2 + 3x + 1 for each xvalue. a. x = 3 b. x = 5 c. x = 2 ...
 6.3.11: The properties you learned in this section involve adding and multi...
 6.3.12: APPLICATION Lara buys a $500 sofa at a furniture store. She buys th...
 6.3.13: Use the distributive property and the properties of exponents to wr...
 6.3.14: Write an equivalent expression in the form a bn. a. 3x 5x3 b. x x5 ...
 6.3.15: Jack Frost started a snowshoveling business. He spent $47 on a new...
 6.3.16: Solve each system. a. b.
Solutions for Chapter 6.3: Multiplication and Exponents
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 6.3: Multiplication and Exponents
Get Full SolutionsChapter 6.3: Multiplication and Exponents includes 16 full stepbystep solutions. Since 16 problems in chapter 6.3: Multiplication and Exponents have been answered, more than 7690 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. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

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 .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(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 .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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 ().

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.