Simplify, if possible.

$$x ^ { 2 } \cdot y ^ { - 3 } \cdot x ^ { - 2 } \cdot y ^ { - 3 }$$

evaluate the expression without using a calculator. cos^-1(- √3/2)

Find te value of x for

$$\log _ { 3 } \left( x ^ { 2 } + 2 x \right) = \log _ { 3 } ( x + 2 )$$

$$F. x=-2,1. G. x=-2. H. x=1. J. no solution$$

There is a nice way to solve nonhomogeneous equation with constant soefficients whose right-hand sides consist of sine or cosine functions:

$$ay''+by'+cy=\begin{cases}R\cos\omega t,\\R\sin\omega t\end{cases}$$

. The idea is to solve a modified equation in which we replace the sine or cosine twrm with the complex exponential $e^{i\omega t}$, $ay''+by'+cy=Re^{i\omega t}$, where the complex constant $i=\sqrt{-1}$ is treated as any real constant. The important fact is that the real solution of $ay''+by'+cy=Re^{i\omega t}$ is the solution of

$$ay''+by'+cy=\begin{cases}R\cos\omega t,\\R\sin\omega t\end{cases}$$

with a cosine on the right-hand side, and the imaginary solution of $ay''+by'+cy=Re^{i\omega t}$ is the solution of

$$ay''+by'+cy=\begin{cases}R\cos\omega t,\\R\sin\omega t\end{cases}$$

with a sine on the right-hand side. Use this idea to solve $y''+25y=6\sin t$.

A function f and a point P are given. Let

$$\theta$$

correspond to the direction of the directional derivative. Write the directional derivative at P as a function of

$$\theta$$

; call this function g.

$$f ( x , y ) = 8 + x ^ { 2 } + 3 y ^ { 2 } ; P ( - 3 , - 1 )$$

Find a vector that points in the same direction as $\mathbf{v}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}$ and has length one-third.

Decide whether you can use the given information to prove that ΔABC ≅ ΔDEF. Explain your reasoning. ∠A ≅ ∠D, ∠B ≅ ∠E, B̅C̅ ≅ E̅F̅

All of the following are generally true about fiber evidence except (a) It is analyzed using microscopes (b) It is possible to determine the composition and quantities of its chemical makeup using gas chromatography and a mass spectrometer (c) It can be used to place a person at a crime scene at a specific time. (d) It is generally considered to be class evidence.

Without a calculator, perform each operation indicated in the expressions below. $\left(\frac{5 x^{2}}{y}\right) \cdot\left(\frac{10}{x}\right)$

Sketch the graph of the equation $y = - 2 ( x + 1 ) ^ { 2 } - 3$ by translating, reflecting, compressing, and stretching the graph of $y = x ^ { 2 }$ appropriately, and then use a graphing utility to confirm that your sketch is correct.

Mrs. Henderson has 16 boys in her class of 24 students. Mr. Gregory has 18 boys in his class of 30 students. Which class has the greater ratio of boys to students? Explain.

Show why there is no solution to the following set.

$$\begin{aligned} x _ { 1 } - 3 x _ { 2 } & = 2 \\ 3 x _ { 1 } + 5 x _ { 2 } & = 7 \\ 5 x _ { 1 } - 2 x _ { 2 } & = - 4 \end{aligned}$$

Write each decimal as a percent. 1.11

Which expression is equal to $\frac { \sin 2 x } { 1 + \cos 2 x } ?$ a. cot x, b. tan x, c. tan 2x, d. cot 2x