Let $p$ and $q$ be distinct primes: Suppose that $H$ is a proper subset of the integers and H is a group under addition that contains exactly three elements of the set $\left\{p, p+q, p q, p^{q}, q^{p}\right\}$. Determine which of the following are the three elements in H.

a. $p q, p^{q}, q^{p}$.

b. $p+q, p q, p^{q}$.

c. $p, p+q, pq$.

d. $p, p^{q}, q^{p}$,

e. $p, p q, p^{q}$

Why is solar energy transferred to Earth by radiation?

A tall cylinder contains 25 cm of water. Oil is carefully poured into the cylinder, where it floats on top of the water, until the total liquid depth is 40 cm. What is the gauge pressure at the bottom of the cylinder?

Janna's health class measures the heart rates of students after they walked for five minutes. The heart rates, in beats per minute, were {102, 74, 86, 74, 96, 95, 103, 102}. Find the mean of the data set. What does the mean tell you? Show your work.

The Hubbert curve predicts that (a) peak oil will occur once half of the supply is used up. (b) finding additional reserves could greatly increase the time to peak oil. (c) natural gas will be cheaper than oil by the early twenty-first century. (d) the cost of oil is independent of the world's oil supply. (e) coal will run out faster than oil.

How do you graph y =1/x²?

Write the coefficient matrix and the augmented matrix of the given system of linear equations.

$$\begin{array}{l}{3 x_{1}+5 x_{2}=8} \\ {2 x_{1}-4 x_{2}=-7}\end{array}$$

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. $f(x)=x^{3}-x^{2}-10 x-8$

In testing for air pollution, a given air sample contained a total of 6.0 parts per million (ppm) of four pollutants, sulfur dioxide (SO2), nitric oxide (NO), nitrogen dioxide (NO2), and carbon monoxide (CO). The ppm of CO was 10 times that of SO2, which in turn equaled those of NO and NO2. There was a total of 0.8 ppm of SO2 and NO. How many ppm of each were present in the air sample?

Find the solution set for the following linear system by calculating the inverse of the coefficient matrix and then using matrix multiplication:

$$\left\{\begin{array}{rr}{4 x_{1}-6 x_{2}+x_{3}=} & {17} \\ {-x_{1}+2 x_{2}-x_{3}=} & {-14} \\ {3 x_{1}-5 x_{2}+x_{3}=} & {23}\end{array}\right.$$

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An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25$^{\circ}$ incline. It rolls, without slipping, to the bottom. a. What is the sphere’s angular velocity at the bottom of the incline? b. What fraction of its kinetic energy is rotational?

Use the quotient rule to simplify the expression. Assume that x > 0.

$$\frac { \sqrt { 72 x ^ { 3 } } } { \sqrt { 8 x } }$$

Suppose that you connect two identical flashlight batteries in series to get a 3 V emf. Write a loop equation to show that the maximum (short-circuit) current these batteries can deliver is exactly the same as the maximum current from one battery.

Find the solution of the given differential equation satisfying the indicated initial condition. y'=4y, y(0)=2