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# Basic Algebra

### Normal subgroups and product of two right cosets

Modern Algebra Group Theory (XXXV) Cosets of Subgroups of a Group Normal Subgroups of a Group A subgroup N of G is a normal subgroup of G if and only if the product of two right cosets of N in G is again a right coset of N in G.

### Radical Signs and Advantages of Rational Exponents; Loudness of Sound - Formula and Measurement; Index of a Sequence - Range, Domain, Aritmetic and Geometric Sequences

1. While the radical symbol is widely used, converting to rational exponents has advantages. Explain an advantage of rational exponents over the radical sign. Include in your answer an example of an equation easier to solve as a rational exponent rather then a radical sign. 2. The loudness of sound is based on intensity l

### The Order of an Element of a group

Modern Algebra Group Theory (XXXIII) Subgroups of a Group The Order of an Element of a group If in the group G, a^5 = e, aba^(-1) = b^2 for a,b belongs to

### Centre of a Group: If G is a group, the centre of G, Z is defined by Z = {z in G|zx = xz, all x in G}. Prove that Z is a subgroup of G. Or Prove that Z is a normal subgroup of G.

Modern Algebra Group Theory (XXX) Subgroups of a Group Centre of a Group If G is

### Normalizer of a Group or Centralizer of a Group: If a is in G define N(a) = {x is in G|xa = ax}.Show that N(a) is a subgroup of G. N(a) is usually called the Normalizer or Centralizer of a in G.

Modern Algebra Group Theory (XXIX) Subgroups of a Group

### Logarithims: Can you tell me how to find log 10000 where log means log to the base of 10 with the answer?

Can you tell me how to find log 10000 where log means log to the base of 10 with the answer? Also, most calculators have 2 different logs on them: log, which is based 10, and ln, which is based e. In computer science, digital computers are based on the binary numbering system where there are only 2 numbers available to the co

### College Algebra: Writing Equations from Word Problems

Please help with the following attached questions: 2,8,12,20 2. If 40L of an acid solution is 75% acid, how much pure acid is there in the mixture? 8. Unknown Numbers: Consider the following problem. The difference between six times a number and 9 is equal to five times the sum of the number and 2. Find the number.

### Logarithm Problems : Change of Base, Graphing and Solving for X

Find the exact values: 1) log (base 10) 1000 2) ln e^-100 3) log (base 5) (1/25) 4) log (base10) (0.1) 5) log (base 12) 3 + log (base 12) 48 6) 2^(log(base 2) 3 + log(base 2) 5) 7) e^(ln 15) 8) e^(3ln2) 9) log(base 8)320 - log(base8)5 -------------------------------------------------------------------------------

1) Using the quadratic equation x2 - 4x - 5 = 0, perform the following tasks: a) Solve by factoring. b) Solve by completing the square. c) Solve by using the quadratic formula. 2) For the function y = x2 - 4x - 5, perform the following tasks: a) Put the function in the form y = a(x - h)2 + k. b) What is the line of s

### Problem Set

(See attached file for full problem description) --- ? Identify the document by typing your full name and section number next to the yellow text. ? Rename the file by adding your last name to current file name (e.g., "u1ip_lastname.doc"). ? Type your answers next to the yellow text. ? To show your work, you will need t

When using the quadratic formula to solve a quadratic equation (ax2 + bx + c = 0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. How can I create three unique equations where the discriminant is positive, zero, or negative, and for each case, explain what this value means to the graph of

### Putting a function in a form y=

Y = x^2 - 4x - 5 = x^2 - 2*2*x - 5 = ( x^2 - 2*2*x +2^2 ) - 2^2 - 5 = (x-2)^2 - 2^2 - 5 = (x-2)^2 - 4 - 5 = (x-2)^2 -9 y = (x-2)^2 -9 How do I put this in the form y=a(x-H)^2+K, how do I graph this function, and why is it not necessary to plot points to graph when using y=a(x-h)^2+K?

### Algebra word problem

If John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be? How do I set up this equation?

Using the Quadratic equation: x2 - 4x - 5 = 0. How do I solve this by factoring? by completing the square? & solve by Quadratic formula?

### For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, h in H. Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

Modern Algebra Group Theory (XXVII) Subgroups of a Group Cosets of Su

### For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, h in H. Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

Modern Algebra Group Theory (XXVII) Subgroups of a Group Cosets of S

### Systems of Inequalities and Finding the Minimum Value : 4x + 3y > 72 and 6x + 10y < 174

The minimum value of z = 5x + 15y, subject to 4x + 3y > 72 6x + 10y < 174 x > 0, y > 0 occurs at: A. (0, 17.4) B. (9, 12) C. (18, 0) D. (29,0)

### Problem 12

Select the point which is in the feasible region of the system of inequalities: 4x + y < 8 2x + 5y < 18 x > 0, y > 0 A. (2,4) B. (-1,2) C. (1,3) D. (4,1)

### Tautologies, The Laws of Logic, The Basic Logical Laws, Absorption Laws (II) : Prove that p disjunction (p conjunction q) <=> p is a tautology

Modern Algebra Logic (XXII) Tautologies The Laws of Logic

### Tautologies, The Laws of Logic, The Basic Logical Laws, Distributive Laws (III) : Prove that p conjunction (q conjunction r) <=> (p conjunction q) conjunction (p conjunction r) is a tautology.

Modern Algebra Logic (XIX) Tautologies The Laws of Logic

### Tautologies, The Basic Logical Laws, Distributive Laws : Prove that p disjunction (q conjunction r) <=> (p disjunction q) conjunction (p disjunction r) is a tautology.

Modern Algebra Logic (XV) Tautologies The Laws of Logic

### Logic, Tautologies, The Basic Logical Laws, The Laws of Addition

Modern Algebra Logic (X) Tautologies The Laws of Logic The Basic Logical Laws The Laws of Addition (I) To prove that p => pVq is a tautology. The fully formatted problem is in the attached file.

### Logarithms, Exponents, Logarithmic Form and Exponential Form (12 Problems)

1. Convert the following equations into logarithmic form: a. 9 = 4^x b. 3 = 6^y c. 5 = 7^y d. X = 9^y 2. Convert the following equations into exponential form: a. X = log36 b. -5 = log3Y c. X= log4Y d. 1000 = log5Z 3. Simplify the following expressions: a. X5 * X7 b. Z10/Z11

### Follow the example and use the least upper bound property of the real numbers to prove that any positive real number has a cube root.

Given an example of squared roots: Let x be a real number such that x > 0. Then there is a positive real number y such that y2 = y?y = x Let S = {s &#1108; R: s>0 and s2<x} The S is not empty since x/2 &#1108; S, if x<2 and 1 &#1108; S otherwise. S is also bounded above since, x+1 is an upper bound for S. Let y be the l

### Speed and Distance

Leon drove 270 miles to the lodge in the same time as Pat drove 330 miles to the lodge. If Pat drove 10 miles per hour faster than Leon, then how fast did each of them drive?

### Distance and Speed

Janet drove 120 miles at x mph before 6:00 a.m. After 6:00 a.m., she increased her speed by 5 mph and drove 195 additional miles. Write a rational expression for her total traveling time. Evaluate the expression for x = 60.

### Prove that if A is a set of positive, finite Lebesgue measure

A point x of a measurable subset A of the reals is called a density point if m( A intersection [x-h, x+h] ) / 2h goes to 1 as h goes to 0 where m is the Lebesgue measure. Prove that if A is a set of positive, finite Lebesgue measure, then almost every point of A is a density point. I would like to note that I can use

### Distribution of union over intersection

Modern Algebra Set Theory (XVIII) Laws of Algebra of Sets

### Context-fee language

Thank you for taking the time to look at my problem. I cannot make math symbols, thus, I will let ^ denote "raised to the power." For example, a^2 is a squared or a "raised to the power" of 2. Also, I will use the symbol * to denote multiplication. For example, 2*7=14. Okay, here is my problem: Show that the language L={ a

### Algebraic Structures, Inverses, Finite Group, Identity

5. Let (A, *) be an algebraic structure, and suppose that A is associative, has an identity, e, and that a &#1028; A has an inverse. Show that if ax = ay, then x = y. 8. Let G be a finite group with identity e, and let . Show that there is an n &#1028; N with a^n = e (Hint: Consider the set {e, a, a2 , ..., am }, where m