 3.1: List the first six powers of: a 2 b 3 c 4
 3.2: Copy and complete the values of these common powers: a 51 = :::: , ...
 3.3: Simplify, then use a calculator to check your answer: a (1)5 b (1)6...
 3.4: Use your calculator to find the value of the following, recording t...
 3.5: Use your calculator to find the values of the following: a 91 b 1 9...
 3.6: Consider 31, 32, 33, 34, 35 .... Look for a pattern and hence find ...
 3.7: What is the last digit of 7217?
 3.8: Write without negative exponents: a ab2 b (ab)2 c (2ab1)2 d (3a2b)2...
 3.9: Write in nonfractional form: a 1 an b 1 bn c 1 32n d an bm e an a2+n
 3.10: Simplify, giving your answers in simplest rational form: a 5 3 0 b ...
 3.11: Write as powers of 2, 3 and/or 5: a 1 9 b 1 16 c 1 125 d 3 5 e 4 27...
 3.12: Read about Nicomachus pattern on page 84 and find the series of odd...
 3.13: Solve for x: a ex = pe b e 1 2 x = 1 e2
 3.14: The current flowing in an electrical circuit t seconds after it is ...
 3.15: Consider the function f(x) = ex. a On the same set of axes, sketch ...
Solutions for Chapter 3: EXPONENTS
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 3: EXPONENTS
Get Full SolutionsChapter 3: EXPONENTS includes 15 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 3: EXPONENTS have been answered, more than 11556 students have viewed full stepbystep solutions from this chapter. Mathematics for the International Student: Mathematics SL was written by and is associated to the ISBN: 9781921972089. This textbook survival guide was created for the textbook: Mathematics for the International Student: Mathematics SL, edition: 3.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.