Write the following numbers using powers-of-ten notation: (a) ten million, (b) sixty thousand, (c) four one-thousandths, (d) thirty-eight billion, (e) your age in months.

Scientific notation is a way to express very large or very small numbers in a concise and standardized form. It consists of two parts: a coefficient and a power of 10. The coefficient is a number greater than or equal to 1 and less than 10, and it's multiplied by 10 raised to an exponent. The exponent indicates how many places the decimal point moves from the original number.

For example, the number 300 can be written in scientific notation as \(3 \times 10^2\), which means that the decimal point in the number 3 is moved two places to the right. Conversely, a small number like 0.005 would be \(5 \times 10^{-3}\) because the decimal moves three places to the left to reach the number 5.

Using the steps in the exercise as a guide, any number can be expressed in scientific notation, making it easier to work with very large or small quantities.

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent indicates how many times the base is multiplied by itself. For example, \(10^3\) means you multiply 10 by itself three times (10 * 10 * 10), which equals 1000.

Properties of Exponentiation:

• \(10^1 = 10\): The exponent of 1 means the base remains the same.
• \(10^0 = 1\): Any non-zero number raised to the power of 0 is 1.
• If the exponent is negative, you take the reciprocal of the base and then apply the positive exponent. For instance, \(10^{-2} = 1/(10^2) = 1/100\).

The steps provided in the textbook solutions use these principles to rewrite numbers as powers of ten, a form of exponentiation.

Mathematical notation is a system of symbols and signs used to denote mathematical objects, operations, relations, and concepts. It's standardized to ensure clarity and consistency across the mathematical community. Notations such as powers-of-ten provide a universal way to express numbers, which is crucial for efficient communication in fields such as science and engineering.

One key aspect of mathematical notation is its ability to condense complex ideas into a simple symbol or equation, as seen when writing large numbers using powers of ten. This method significantly reduces the complexity of operations and improves the readability of mathematical expressions.

The term 'order of magnitude' refers to the scale or size of a quantity, typically measured in powers of ten. It is used to compare and describe the relative magnitude of numbers. When two numbers differ by one order of magnitude, one is roughly ten times greater or smaller than the other.

For instance, the difference between 1,000 (\(10^3\)) and 10,000 (\(10^4\)) is one order of magnitude. Orders of magnitude are particularly useful in scientific contexts, where they help gauge the significance of numbers without requiring precise values.

In the textbook exercises, understanding the order of magnitude can aid students in rapidly assessing the size of different numbers and assisting in solving problems related to relative scales and sizes.