Simplify using the Laws of Exponents.

$$\left(m^{8}\right)^{5}=$$

Solve the equation. Check your solution. -17 = 6p - 5

Determine whether the vectors v and w are parallel. If the vectors are parallel, are their directions the same or opposite? $\mathbf{v}=\langle\sqrt{2}, 1,3\rangle, \quad \mathbf{w}=\langle 2,1,9\rangle$

Find the indefinite integrals. $\int(5 \cos x-3 \sin x) d x$

Evaluate the expression. Express the result in scientific notation.

$$\frac{9.45 \times 10^{10}}{1.5 \times 10^{6}}=$$

The demand for a product is given, for p, $q \geq 0$, by $p=f(q)=50-0.03 q^{2}$. Find the p- and q-intercepts for this function and interpret them in terms of demand for this product.

Polar coordinates of a point are given. Find other polar coordinates $(r, \theta)$ of the point for which: (a) $r>0,-2 \pi \leq \theta<0$. (b) $r<0,0 \leq \theta<2 \pi$. (c) $r>0,2 \pi \leq \theta<4 \pi$. $\left(1, \frac{\pi}{2}\right)$

$$- 4 ( x - 5 ) =$$

Verify the following formulas for every positive integer n:

$$\frac{d^{n}}{d x^{n}} \cos (a x)=a^{n} \cos \left(a x+\frac{n \pi}{2}\right)$$

A random variable X has a continuous distribution function F, where

$$F(t)=\left\{\begin{array}{lll} {0} & {\text { if }} & {t \leq 0} \\ {c t} & {\text { if }} & {0 \leq t \leq 1} \\ {1} & {\text { if }} & {t>1} \end{array}\right.$$

(a) Determine the constant c and describe the density function f. (b) Compute the probabilities $P\left(X=\frac{1}{3}\right), P\left(X<\frac{1}{3}\right), P\left(|X|<\frac{1}{3}\right)$

Use Green’s Theorem to evaluate the line integral $\int_{C}(y-x) d x+(2 x-y) d y$ for the given path. C: boundary of the region lying inside the rectangle with vertices (5, 3), (−5, 3), (−5, −3), and (5, −3), and outside the square with vertices (1, 1), (−1, 1), (−1, −1), and (1, -1)

Indicate whether the statement is true or false. The derivative of $f(t)=e^{t^{2}}$ is $f^{\prime}(t)=2 e^{2 t}$.

Show that the moments of inertia about the coordinate axes are $I_{x}=I_{x y}+I_{x z}, \ I_{y}=I_{y x}+I_{y z}, \ I_{z}=I_{z x}+I_{z y}.$

A function f is defined by the power series $f(x)=\sum_{k=0}^{\infty}(-1)^{k}(x+3)^{k}$. (a) Find the domain of f. (b) Evaluate f(-3) and f(-2.5). (c) Find f by summing the series.