Use a radical symbol to denote and evaluate the principal nth root (if it exists) of the number. Then find any other real nth roots of the number.

$$\begin{matrix} \text{Number} & \text{n}\\ \text{27} & \text{3}\\ \end{matrix}$$

You are researching the price of MP3 players. You found an average price of $58.80. One MP3 player costs$56 and another costs $62. Find the price of the third MP3 player.

Invitations are sold in packs of 3. Emily needs 77 invitations for her party. How many packs of invitations does she need?

The essential elements of a hand winch are shown. The toothed wheel and the handle OA pivot independently about the same axis through O. When bearing against the radial flank of a tooth, the pawl BC locks at the $20^{\circ}$ angle shown. When separated from the tooth flank, the pawl can rotate clockwise relative to OA to allow clockwise rotation of OA relative to the winch drum in preparation for another pull. A lock not shown prevents clockwise rotation of the toothed wheel during repositioning of the handle. If P = 400 N, determine the tensions ${T}_{{1}}$ and ${T}_{{2}}$. Neglect friction at the base of the winch.

A 5-mm-thick stainless steel strip $(k=21 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},$ $\left.\rho=8000 \mathrm{kg} / \mathrm{m}^{3}, \text { and } c_{p}=570 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\right)$ is being heat treated as it moves through a furnace at a speed of 1 cm/s. The air temperature in the furnace is maintained at $900^{\circ} \mathrm{C}$ ith a convection heat transfer coefficient of $80 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}.$ If the furnace length is 3 m and the stainless steel strip enters it at $20^{\circ} \mathrm{C},$ determine the temperature of the strip as it exits the furnace.

Perform the indicated operation(s) and write the result in standard form. $(5+6 i)^{2}$

The blades of the portable fan generate a 1.2-lb thrust T as shown. Compute the moment $M_O$ of this force about the rear support point O. For comparison, determine the moment about O due to the weight of the motor-fan unit AB, whose weight of 9 lb acts at G.

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Suppose C(r) is the total cost of paying off a car loan borrowed at an annual interest rate of r%. What are the units of C'(r)? What is the practical meaning of C'(r)? What is its sign?

Find the zeros of the polynomials. $P(z)=z^{2}-2 i z-1$

Prepare a poster for the CEO detailing your findings from your quadratic function investigation. Include any insights you and your teammates found. Explain your conclusions and justify your statements. Remember to include both a table and a complete graph of your parabola with all special points carefully labeled. Be thorough and complete.

Find the general solution on the interval $(0, \infty)$ of the given DE using variation of parameters. $y^{\prime \prime}+4 y^{\prime}+4 y=\frac{e^{-2 x}}{x^{2}}$

Suppose f is twice differentiable on an interval I (i.e. , f " exists on I). Suppose that the points 0 and 2 belong to I and that $f(0)=f(1)=0 \text { and } f(2)=1$. Prove that

$f^{\prime}(a)=\frac{1}{2}$ for some point a in I

Show that the velocity field of a line source of strength 2m can be found by integrating the (three-dimensional) velocity field of a point source of strength m dz at (0, 0, z) over the whole z-axis. Why doe s the integral correspond to a line source of strength 2m rather than strength m? Can the potential of the line source be obtained by integrating the potentials of the point sources?