In many applications, it is necessary to write expressions in the form $c x^n$, where $c$ is a constant and $n$ is an integer. Write the expressions in this form.

$$\frac{1}{4 x^2}$$

Factor completely.

$$z^2-6 z k-16 k^2$$

Determine which statements are true and which are false. Prove those that are true and disprove those that are false.

$$\sqrt { 2 } / 6$$

is rational.

In many applications, it is necessary to write expressions in the form $c x^n$, where $c$ is a constant and $n$ is an integer. Write the expressions in this form.

$$\frac{1}{x^2}$$

Briefly explain the importance of each of the following. culture

Find the sum of each arithmetic series. The first 100 terms of the series

$$100 + 98 + 96 + 94 + \cdots$$

Complete the following statements about your family's car or a favorite car, using as many descriptive words as possible. También tiene _______

Assuming that $n$ is an even perfect number, say $n=2^{k-1}\left(2^{k}-1\right),$ prove that the product of the positive divisors of $n$ is equal to $n^{k} ;$ in symbols,

$$\prod_{d | n} d=n^{k}$$

Solve the inequality.

$$2 ( k - 5 ) < 2 k + 5$$

Identify the property that justifies each statement. $x=y, \text { so } y=x$

Read each sentence. If the sentence is true, write YES. If the sentence is not true, write NO and explain why.

Ironclads were of little importance in the war at sea.

Convert each of the following decimal numbers to its binary equivalent.

(a) 24

(b) 91

(c) 135

(d) 396

Sketch the graph of the function. Show vertical and horizontal asymptotes and x-intercepts.

$$y=\frac{x^2-16}{x^2-6 x+8}$$

Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs $\$ 14,000 /$ day to operate, and it yields $50 \mathrm{oz}$ of gold and $3000 \mathrm{oz}$ of silver per day. The Horseshoe Mine costs $\$ 16,000 /$ day to operate, and it yields $75 \mathrm{oz}$ of gold and 1000 ounces of silver per day. Company management has set a target of at least $650 \mathrm{oz}$ of gold and 18,000 oz of silver.

a. How many days should each mine be operated so that the target can be met at a minimum cost?

b. Find the range of values that the Saddle Mine's daily operating cost can assume without changing the optimal solution.

c. Find the range of values that the requirement for gold can assume.

d. Find the shadow price for the requirement for gold.