Use the polar form to find each power. Write the power in rectangular form.

$$(\sqrt{3}-i)^{3}$$

Use integration by parts to evaluate the following integrals.

$$\int x \sinh x d x$$

Calculate the derivative.

$$y = \tanh ^ { - 1 } ( \ln t )$$

Show that: a. ln 27 = 3 ln 3 b. $\ln \sqrt{3}=\frac{1}{2} \ln 3$ c. $\ln \left(\frac{1}{16}\right)=-4 \ln 2$ d. $\ln \left(\frac{1}{6}\right)=-\ln 6$ e. $\ln \left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{2} \ln 2$ f. $\ln \left(\frac{e}{5}\right)=1-\ln 5$ g. $\ln \sqrt[3]{5}=\frac{1}{3} \ln 5$ h. $\ln \left(\frac{1}{32}\right)=-5 \ln 2$ i. $\ln \left(\frac{1}{\sqrt[5]{2}}\right)=-\frac{1}{5} \ln 2$

If the discriminant is zero, how many times does the graph of a quadratic equation intersect the x-axis? A. 0 B. 1 C. 2 D. More than twice

Multiply.

$$\left( r - 4 t ^ { 4 } \right) \left( r - 4 t ^ { 4 } \right)$$

Determine the domain and find the derivative of $f(x)=\ln|\dfrac{x+2}{x^3-1}|$

Which of the following correctly lists types of organisms in aquatic ecosystems from shallowest to deepest? A. plankton, nekton, benthos B. plankton, benthos, nekton C. benthos, plankton, nekton D. benthos, nekton, plankton

A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is $10$ m/s. It strikes the ground with a speed of $60$ m/s. How tall is the building?

Determine whether the statement is true or false. Explain. Most tiles on floors are laid out with a pattern using tessellations.

Graph each function. y = -7x^2

Write a program that asks the user to enter five test scores. The program should display a letter grade for each score and the average test score. Write the following functions in the program:

$$\begin{matrix} • \text{calc}\_\text{average}\text{ —This function should accept five test scores as arguments and}\\\text{ return the average of the scores.}\\ • \text{determine}\_\text{grade —This function should accept a test score as an argument}\\\text{ and return a letter grade for the score based on the following grading scale:}\\ \end{matrix}$$

$$\begin{matrix} \text{Score} & \text{Letter Grade}\\ \hline \text{90–100} & \text{A}\\ \text{80–89} & \text{B}\\ \text{70–79} & \text{C}\\ \text{60–69} & \text{D}\\ \text{Below 60} & \text{F}\\ \end{matrix}$$

Simplify each expression.

$$\frac { a ^ { 2 } x - b ^ { 2 } x } { b y - a y }$$

Carmen and Judi dock a canoe. 80.0-kg Carmen moves forward at 4.0 m/s as she leaves the canoe. At what speed and in what direction do the canoe and Judi move if their combined mass is 115 kg?