Simplify the expression.

$$\left( a ^ { 2 } \right) ^ { - 3 } \left( a ^ { 3 } b \right) ^ { 2 } \left( b ^ { 3 } \right) ^ { 4 }$$

Suppose you are given that a = 5 and 4a + b = 6. What can you prove by using these statements and the Substitution Property?

Find 1834 - 712 + 78.

Use the quadratic formula to solve the equation. Approximate the solution to the nearest hundredth, when appropriate.

$$0.01 x ^ { 2 } = - 0.08 x - 0.15$$

Show that the nth Taylor polynomial for sinh x about x=ln 4 is

$$\sum _ { k = 0 } ^ { n } \frac { 16 - ( - 1 ) ^ { k } } { 8 k ! } ( x - \ln 4 ) ^ { k }$$

Rewrite the formula for the area of a rhombus for the special case of a square with side length s. Show that this is the same as the formula for the area of a square, A = s².

Simplify.

$$\ln \sqrt { e }$$

$$\text{ Consider the following joint probability density function of the random variables } X \text{ and } Y :$$

$$f(x,y)=\frac{3x-y}{9}, \text{ for } 1<x<3, \text{ } 1<y<2,$$

$$f(x,y)=0, \text{ elsewhere.}$$

[

$$(a) \text{ Find the marginal density functions of } X \text{ and } Y .$$

$$(b) \text{ Are } X \text{ and } Y \text{ independent?}$$

$$(c) \text{ Find } P(X >2).$$

An equation of an ellipse is given. Sketch a graph of the ellipse. $\frac{x^{2}}{4}+\frac{y^{2}}{36}=1$

What are three strategies people often use to solve problems? Explain how you have used one of these strategies to solve a problem.

A hockey puck moving at 0.45 m/s collides with another puck that was at rest. The pucks have equal mass. The first puck is deflected

$$37 ^ { \circ }$$

to the right and moves off at 0.36 m/s. Find the speed and direction of the second puck after the collision.

Write each exponential expression as a decimal.

$$10^2$$

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What is the area of a circle with a diameter of 10 meters? A. 10

$$\left. \begin{array} { l } { \text { 71. Let } c \text { be a scalar, a and b be vectors, and } \mathbf { X } = \langle x , y , z \rangle . \text { Show } } \\ { \text { that the equation } ( \mathbf { X } - \mathbf { a } ) \cdot ( \mathbf { X } - \mathbf { b } ) = c ^ { 2 } \text { defines a sphere with center } }\mathbf { m } = \frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } ) \text { and radius } R ,\\ \text { where } R ^ { 2 } = c ^ { 2 } + \left\| \frac { 1 } { 2 } ( \mathbf { a } - \mathbf { b } ) \right\| ^ { 2 } \end{array} \right.$$