Evaluate the product. Express the result in scientific notation.

$$(7.5 \times 10^6)(2.5 \times 10^{-2})$$

Find the characteristic of the given ring.

$$ℤ_3× ℤ_4$$

Graph the following functions by making a table of values. $f(x)=x^{5}-3 x^{4}+9 x^{3}-x^{2}+2$

Consider the following reaction :

$$2 \mathrm { Mg } ( s ) + \mathrm { O } _ { 2 } ( g ) \longrightarrow 2 \mathrm { MgO } ( s ) \ \Delta H = - 1204 \mathrm { kJ }$$

Calculate the amount of heat transferred when 3.55 g of Mg(s) reacts at constant pressure.

For a recent period of 100 years, there were 93 major earthquakes (at least 6.0 on the Richter scale) in the world (based on data from the World Almanac and Book of Facts). Assuming that the Poisson distribution is a suitable model, find the mean number of major earthquakes per year, then find the probability that the number of earthquakes in a randomly selected year is 2.

Deduce that

$$\int _ { 0 } ^ { \pi / 6 } \cos \left( x ^ { 2 } \right) d x \geq \dfrac { 1 } { 2 }$$

What are the three components of ATP?

How does the role of B cells in the immune response differ from that of helper T cells?

Use a calculator. Round to the nearest tenth of a degree. Given

$$\cos \theta = 0.4756$$

, find

$$\theta$$

.

Rectangle ABCD has a width of 8 yards and a length of 20 yards. Rectangle QRST, which is similar to rectangle ABCD, has a length of 40 yards. Find the scale factor of rectangle ABCD to rectangle QRST and the perimeter of each rectangle.

Find the absolute extrema of the function over the region R. (In each case, contains the boundaries.) f(x, y) = (2x - y)² R: The triangular region in the plane with vertices (2, 0), (0, 1), and (1, 2)

An ideal gas is heated at a constant pressure of

$$2.0 \times 10 ^ { 5 }$$

Pa from a temperature of

$$- 73 ^ { \circ } \mathrm { C }$$

to temperature of

$$+ 27 ^ { \circ } \mathrm { C }$$

. The initial volume of the gas is 0.10

$$m^3$$

. The heat energy supplied to the gas in this process is 25 kJ. What is the increase in internal energy of the gas?

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. For every vector

$$\mathbf { b } \text { in } \mathbb { R } ^ { m },$$

applying the projection matrix P for the linear system

$$A \mathbf { x } = \mathbf { b }$$

to the vector

$$\mathbf b$$

results in a vector that lies in colspace(A).

Consider the sequence 0, 1, 3, 6, 10, 15, 21, ... (a) List the next two terms of the sequence. (b) Assuming the sequence is denoted by

$$A _ { 1 } , A _ { 2 } , A _ { 3 } , \dots$$

, give an explicit formula for

$$A_N$$

, (c) Assuming the sequence is denoted by

$$P _ { 0 } , P _ { 1 } , P _ { 2 } , \dots$$

, give an explicit formula for

$$P_N$$