 10.1: Find each square root. If necessary, round to the nearest hundredth.
 10.2: Find each square root. If necessary, round to the nearest hundredth.
 10.3: Find each square root. If necessary, round to the nearest hundredth.
 10.4: PAINTING Todd is painting a square mural with an area of 81 square ...
 10.5: Simplify each expression.
 10.6: Simplify each expression.
 10.7: Simplify each expression.
 10.8: Simplify each expression.
 10.9: Solve each equation.
 10.10: Solve each equation.
 10.11: GEOMETRY The triangle has an area of 120 square centimeters. Find h.
 10.12: Use cross products to determine whether each pair of ratios forms a...
 10.13: Use cross products to determine whether each pair of ratios forms a...
 10.14: Use cross products to determine whether each pair of ratios forms a...
 10.15: MODELS A collectors model train is scaled so that 1 inch on the mod...
 10.16: Solve each equation. Check your solution.
 10.17: Solve each equation. Check your solution.
 10.18: MULTIPLE CHOICE The surface area S of a cone can be found by using ...
 10.19: PHYSICS When an object is dropped from the top of a 250foot tall b...
 10.20: SKYDIVING The approximate time t in seconds that it takes an object...
 10.21: Find each product.
 10.22: Find each product.
 10.23: MOTION The velocity of a dropped object can be found using v = 2gd ...
 10.24: Solve each equation. Check your solution.
 10.25: Solve each equation. Check your solution.
 10.26: Solve each equation. Check your solution.
 10.27: Solve each equation. Check your solution.
 10.28: Solve each equation. Check your solution.
 10.29: Solve each equation. Check your solution.
 10.30: FREE FALL Assuming no air resistance, the time t in seconds that it...
 10.31: If c is the measure of the hypotenuse of a right triangle, find eac...
 10.32: If c is the measure of the hypotenuse of a right triangle, find eac...
 10.33: If c is the measure of the hypotenuse of a right triangle, find eac...
 10.34: If c is the measure of the hypotenuse of a right triangle, find eac...
 10.35: If c is the measure of the hypotenuse of a right triangle, find eac...
 10.36: If c is the measure of the hypotenuse of a right triangle, find eac...
 10.37: Determine whether the following side measures form right triangles.
 10.38: Determine whether the following side measures form right triangles.
 10.39: Determine whether the following side measures form right triangles.
 10.40: Determine whether the following side measures form right triangles.
 10.41: MOVING The door of Julios apartment measures 7 feet high and 3 feet...
 10.42: Find the distance between each pair of points with the given coordi...
 10.43: Find the distance between each pair of points with the given coordi...
 10.44: Find the distance between each pair of points with the given coordi...
 10.45: Find the distance between each pair of points with the given coordi...
 10.46: Find the value of a if the points with the given coordinates are th...
 10.47: Find the value of a if the points with the given coordinates are th...
 10.48: SAILING A boat leaves the harbor and sails 5 miles east and 3 miles...
 10.49: For each set of measures given, find the measures of the remaining ...
 10.50: For each set of measures given, find the measures of the remaining ...
 10.51: For each set of measures given, find the measures of the remaining ...
 10.52: For each set of measures given, find the measures of the remaining ...
 10.53: HOUSES Josh plans to make a model of his house in the scale 1 inch ...
Solutions for Chapter 10: Radical Expressions and Triangles
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 10: Radical Expressions and Triangles
Get Full SolutionsAlgebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 10: Radical Expressions and Triangles includes 53 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 53 problems in chapter 10: Radical Expressions and Triangles have been answered, more than 37165 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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).

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.

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.

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

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 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.