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,

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

Why is it important to only change one experimental variable at a time in an experiment?

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

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

Complete the following conversations that you overhear in school.
A: ¿Qué hay... la mochila? B: Hay muchos libros.

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

Determine whether each of the statements is true or false, and explain why. In a fair game, the elements of the payoff matrix must sum to 0.

If you have a friend in an abusive relationship, you may be tempted to "rescue" her or him. You may try to persuade your friend to leave the relationship by criticizing the abuser. However, criticizing the abuser may simply make the victim less willing to confide in you. Ask your friend how she or he feels. Express your concerns in a way that focuses on the abusive behavior rather than on the abuser. You might say, "I'm worried that you're getting hurt," rather than "That jerk doesn't deserve you." Let your friend know that she or he can always count on you for sympathy and support. That way, when your friend does feel ready to leave the relationship, she or he will be more likely to turn to you for help. The purpose of this piece is (a) to describe the consequences of abuse. (b) to discuss ways to prevent abuse. (c) to let teens know how to help a friend in an abusive relationship. (d) to list sources of help for abused teens.

All-purpose fertilizers contain the essential elements nitrogen, phosphorus, and potassium. A typical fertilizer carries numbers on its label, such as "5-10-5". These numbers represent the $\% \mathrm{~N}$, $\% \mathrm{P}_2 \mathrm{O}_5$, and $\% \mathrm{~K}_2 \mathrm{O}$, respectively. The $\mathrm{N}$ is contained in the form of a nitrogen compound, such as $\left(\mathrm{NH}_4\right)_2 \mathrm{SO}_4, \mathrm{NH}_4 \mathrm{NO}_3$, or $\mathrm{CO}\left(\mathrm{NH}_2\right)_2$ (urea). The $\mathrm{P}$ is generally present as phosphate, and the $\mathrm{K}$ as $\mathrm{KCl}$. The expressions $\% \mathrm{P}_2 \mathrm{O}_5$ and $\% \mathrm{~K}_2 \mathrm{O}$ were devised in the nineteenth century, before the nature of chemical compounds was fully understood. To convert from $\% \mathrm{P}_2 \mathrm{O}_5$ to $\mathrm{P}_2 \mathrm{O}_5$ and $2 \mathrm{~mol} \mathrm{~K} / \mathrm{mol} \mathrm{K} / \mathrm{O}$ must be used, $2 \mathrm{~mol} / \mathrm{mol}$ with molar masses. Assuming three-significant-figure precision, what is the percent composition of the "5-10-5" fertilizer in $\% \mathrm{~N}, \% \mathrm{P}$, and $\% \mathrm{~K}$ ?

Find the Maclaurin series for the function. (Use the table of power series for elementary functions.) \

$$g(x)=2 \sin x^3$$

Circle the word or words modified by the adverb in italics. On the blank, write $v$ if the adverb modifies a verb. Write $adj$. if the adverb modifies an adjective. Write $adv$. if the adverb modifies another adverb.\ _____Tuyen was completely calm when we jumped out and yelled “Happy Birthday!”

Find the sum.

$$\sum _ { k = 2 } ^ { 5 } ( 5 k )$$

Use your results above to complete these statements: The quotient of two integers with the same sign is ______. The quotient of two integers with different signs is ______.

The vectors $(8,6)$ and $(12, k)$ are parallel. Find the value of $k$.

The vectors $(8, k)$ and $(9,6)$ are perpendicular. Find the value of $k$.