Let ?∈ℕ, and assume √? is irrational. Show that ℚ(√?)={?+?√?∶?,?∈ℚ} is a field (show that there is multiplicative commutativity and multiplicative inverse). What would change if ℚ was replaced with ℝ.

Exercise 10-69 (Algorithmic)
Stock Dividends

Penguin Company has the following information regarding its common stock:

Its common stock is $15 par, with 300,000 shares authorized, 132,000 shares issued, and 130,600 shares outstanding.

On August 22, 2013, Penguin declared and paid a 15% stock dividend when the market price of the common stock was $28 per share.

1. Prepare the journal entry to record declaration and payment of this stock dividend. For a compound transaction, if an amount box does not require an entry, leave it blank.

2. Prepare the journal entry to record declaration and payment assuming it was a 30% stock dividend.

Daryl Kearns saved $240,000 during the 25 years that he worked for a major corporation. Now he has retired at the age of 50 and has begun to draw a comfortable pension check every month. He wants to ensure the financial security of his retirement by investing his savings wisely and is currently considering two investment opportunities. Both investments require an initial payment of $188,500. The following table presents the estimated cash inflows for the two alternatives: Year 1 Year 2 Year 3 Year 4 Opportunity #1 $ 55,675 $ 58,780 $ 78,870 $ 101,420 Opportunity #2 103,500 109,400 17,800 14,100 Mr. Kearns decides to use his past average return on mutual fund investments as the discount rate; it is 9 percent. (PV of $1 and PVA of $1) (Use appropriate factor(s) from the tables provided.) Required Compute the net present value of each opportunity. Which should Mr. Kearns adopt based on the net present value approach? Compute the payback period for each opportunity. Which should Mr. Kearns adopt based on the payback approach? Compute the net present value of each opportunity. Which should Mr. Kearns adopt based on the net present value approach? (Round your intermediate calculations and final answer to two decimal places.) Net Present Value Opportunity 1 Opportunity 2 Which opportunity should be chosen? Payback Period Opportunity 1 Opportunity 2 Which opportunity should be chosen?

Prove that every subgroup of Q₈ is normal. For each subgroup, find the isomorphism type of the corresponding quotient.

I have n books with n different titles. I want to put them on the shelves in my library. How many different ways are there to arrange them on one shelf? On two shelves? On k shelves?

This question is about using Principle of Inclusion-Exclusion Formula, and has to be used to solve this problem:

How many arrangements of a, a, a, b, b, b, c, c, c have no adjacent letters the same? (Hint: This is tricky—not a normal inclusion–exclusion problem.)

Find the infinite series solution about x = 0 for the following differential equation x2y"+ 4xy' + (2+x)y = 0,without using k substitution and using Bessel's, Legrende's, or frobenius equations.

Use the table of values to estimate S ^6 0 f(x) dx, use three equal subintervals and the

(only solve part D)

1. Left end points
2. Right end points
3. Midpoints

When f is an increasing function, how does each estimate compare with the actual value? Explain your reasoning

D. Also find L6, R6, T3, and T6. Remember that Tn is the average of Ln and Rn. Finally, use your calculator to do quadratic regression to find a decent model for your data [let’s call it q(x)] and have your calculator approximate S ^6 0 q (x) dx

Solving Differential Equations using Laplace Transform

a) y" - y' -2y = 0

y(0) = 1, y'(0) = 0

answer: y = 1/3e^2t + 2/3e^-t

b) y" + y = sin2t

y(0) = 2, y'(0) = 1

answer: y(t) = 2cost + 5/3 sint - 1/3sin2t

c) y^4 - y = 0

y(0) = 0, y'(0) = 1, y"(0) = 0, y'''(0) = 0

answer y(t) = (sinht + sint)/2

Find the matrix P that diagonalizes A, and check your work by computer P^-1AP. This matrix is

[-14 12]

[-20 17]

I've tried this problem, and I keep getting the eigenvalues of λ=1, 2 and the eigenspace [4 5] for λ=1, and eigenspace [3 4] for λ=2. However, whenever I check it with P^-1AP, it doesn't produce a diagonal matrix.

A supplier to a music store buys compact discs at $1 per unit and sells them to the music store at $5 per unit. The music store sells each disc to the end consumer at $10. At the retail price, the market demand distribution is give as follows

1. With no contract in place, determine how much the music store should order to maximize it’s expected profit? (8 points)

1. (4 points) What is the total expected profit of the supply chain corresponding to the order quantity calculated from question 1? Show the calculations for the expected profit at the music store and supplier.

2. Now, the supplier to the music store agrees to buyback discs that have not sold at $2.50 per disc. What is the total expected profit of the supply chain if the music store orders 4 CDs? Show the calculations for the expected profit at the music store and supplier. (7 points)

3. The supplier agrees to sell each disc to the music store at $1 instead of $5. The music store agrees to share 35 % of the revenue from each disc sold. What is the total expected profit of the supply chain if the music store orders 2 CDs? Show the calculations for the expected profit at the music store and supplier. (6 points)

Let S = {1,2,3,...,10}.

a. Find the number of subsets of S that contain the number 5.

b. Find the number of subsets of S that contain neither 5 nor 6.

c. Find the number of subsets of S that contain both 5 and 6.

d. Find the number of subsets of S that contain no odd numbers.

e. Find the number of subsets of S that contain exactly three elements.

f. Find the number of subsets of S that contain exactly three elements, one of which is 3.

g. Find the number of subsets of S that contain exactly five elements, all of them even.

h. Find the number of subsets of S with exactly five elements, including 3 or 4 but not both.

2. Guyana forestry Commission wishes to carry out a survey to characterize Acai berry palms within Guyana.

(a) Identify the target population.

(b) Suggest and justify an appropriate sampling technique

(c) Suggest and justify an appropriate sample size.

Emery Pharmaceutical uses an unstable chemical compound that must be

kept in an environment where both temperature and humidity can be controlled.

Emery uses 200 pounds per month of the chemical, estimates the holding cost to be

£3.33 (because of spoilage), and estimates order costs to be £10 per order. The cost

schedules of two suppliers are as follows:

Vendor 1 Vendor 2 Quantity Price/LB (£) Quantity Price/LB (£) 1-49 35.00 1-74 34.75 50-74 34.75 75-149 34.00 75-149 33.55 150-299 32.80 150-299 32.35 300-499 31.60 300-499 31.15 500+ 30.50

500+ 30.75

Vendor 3 Vendor 4 Quantity Price/LB (£) Quantity Price/LB (£) 1-99 34.50 1-199 34.25 100-199 33.75 200-399 33.00 200-399 32.50 400+ 31.00

400+ 31.10

a) What quantity should be ordered, and which supplier should be used?

b) Discuss factor(s) should be considered besides total cost.

Please with explanations

Show that the groups of the following orders have a normal Sylow subgroup.

(a) |G| = pq where p and q are primes.

(b) |G| = p^aq where p and q are primes and q < p.

(c) |G| = 4p where p is a prime greater than four.