Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. $3 \cdot 4^{x}+4 \cdot 2^{x}+8=0$

Is it possible for two events A and B to be both mutually exclusive and independent? Justify your answer.

Dietary proteins must be denatured for efficient digestion. Why?

Is the following function linearly independent on the given interval?

$e^{a x}, e^{-a x}$ (any interval)

What is the unexpected result of Krakauer's request of Harris?

Find the sum of the infinite geometric series, if it exists. $20+2+0.2+0.02+0.002+\cdots$

Most mitochondrial and chloroplast proteins are made _______.

A normal shock wave exists in an airflow. The absolute pressure, velocity, and temperature just upstream from the wave are 207 kPa, 610 m/s, and −17.8$^{\circ} \mathrm{C}$, respectively. Calculate the pressure, velocity, temperature, and sonic velocity just downstream from the shock wave.

Evaluate each limit, if it exists

$$\lim _{x \rightarrow 1} \frac{\sqrt{x}-x^2}{1-\sqrt{x}}$$

Fill in the blanks with an appropriate word, phrase, or symbol(s). A table that summarizes the results of an election is called a(n) _ table.

A computer chip dissipates 2 kJ of electric work over time and rejects that work as heat transfer from its 50$^{\circ} \mathrm{C}$ surface to 25$^{\circ} \mathrm{C}$ air. How much entropy is generated in the chip? How much entropy, if any, is generated outside the chip?

Find the average rate of change of the function over a given interval. Give a decimal approximation rounded to e decimal places.

$$P(t)=4.7 \ln t+1.8, \quad[16,84]$$

The time required to complete a job varies inversely as the number of people working on the job. It takes 8 people 12 days to do a job. a. How many days will it take if 15 people work on the job? b. If the contractor wants to complete the job in 8 days, how many people should work on the job?

A large tank containing air at $125 \mathrm{psia}$ and $175^{\circ} \mathrm{F}$ is attached to a converging-diverging nozzle that has a throat area of $1.5$ in $^2$ through which the air is exiting. A normal shock sits at a point in the nozzle where the area is $2.5 \mathrm{in}^2$. The nozzle exit area is $3.5 \mathrm{in}^2$. What are the Mach numbers just after the shock and at the exit? What are the stagnation and static pressures before and after the shock?