Draw structures that show the similarity between the decarboxylation of the pyruvate–TPP intermediate and the decarboxylation of a β-keto acid.

Find each sum. $2 + 4 + 6 + ... + 2n$

Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible.

Given that P(A)=0.3, P(B)=0.5, and P(A and B)=0.15, use the formula

$$P \left( E _ { 2 } | E _ { 1 } \right) = \frac { P \left( E _ { 1 } \text { and } E _ { 2 } \right) } { P \left( E _ { 1 } \right) }$$

to determine (a) P(A|B), (b) P(B|A), (c) Are A and B independent? Explain.

If

$$\sin \theta = - \frac { 5 } { 13 }$$

and cos

$$\theta$$

0, find the following:

$$\tan 2 \theta$$

In what type of economic analysis do questions have a "right" or "wrong" answer? In what type of economic analysis do questions not necessarily have a "right" answer? On what type of economic analysis do economists tend to disagree most frequently? Why might economists disagree? Explain.

Colby measured the circumference and diameter of four circles. Then he divided the circumference by the diameter of each circle to find the number of diameters in a circumference. Here are his answers: 3.12, 3.2, 3.15, 3.1. Find the average of Colby's answers. Round the average to the nearest hundredth.

Graph the system of linear inequalities and indicate the solution set.

$$\left. \begin{array} { l } { x - y > 5 } \\ { 6 x + 5 y \leq 30 } \end{array} \right.$$

write the polynomial as the product of linear factors and list all the zeros of the function. g(x) = x^2 + 10x + 17

Identify the plot, write an equation, and solve the problem. A whole pizza was cut into eights. If Hector eats

$$\frac { 3 } { 8 }$$

of the pizza and Rita eats

$$\frac { 1 } { 4 }$$

of the pizza, then what fraction of the pizza remains?

Write each set in roster form and in set-builder notation. N is the set of real numbers that are factors of 12.

Solve each compound inequality. Graph your solutions.

$$- 4 < k + 3 < 8$$

Draft tubes are often installed at the exit of Kaplan and Francis turbines. Explain why such draft tubes are advantageous.

Estimate the square root to the nearest integer. Do not use a calculator.

$$- \sqrt { 139 }$$