Question

For each of the following descriptions, identify the power of 10 being indicated by a prefix in the measurement. a. For my lunch, I had a cheeseburger that contained 1/8 of a kilogram of beef. b. The sign on the highway I just passed said "Toronto 50 kilometers." c. The hard drive on my new computer has a storage capacity of 40 gigabytes. d. My favorite radio station broadcasts at 93.7 megahertz on the FM dial. e. The liquid medication I have to give my dog says it contains 2 milligrams of active ingredient per cubic centimeter. f. I just won the lottery for "megabucks"!

Short answer

a. \(0.125 \times 10^3\) grams b. \(50 \times 10^3\) meters c. \(40 \times 10^9\) bytes d. \(93.7 \times 10^6\) hertz e. \(2 \times 10^{-3}\) grams per cubic centimeter f. \(10^6\) bucks

Step-by-step solution

a. Kilogram (

In the metric system, the prefix "kilo" (symbol: k) indicates a factor of 10^3, which means we have a factor of 1,000. Therefore, 1/8 of a kilogram is equal to 0.125 of a kilogram, or \(0.125 \times 10^3\) grams.

b. Kilometers (

Just as in part a, the prefix "kilo" (symbol: k) here indicates a factor of 10^3. Therefore, 50 kilometers is equal to \(50 \times 10^3\) meters.

c. Gigabytes (

In this context, the prefix "giga" (symbol: G) indicates a factor of 10^9. Therefore, a storage capacity of 40 gigabytes is equal to \(40 \times 10^9\) bytes.

d. Megahertz (

The prefix "mega" (symbol: M) indicates a factor of 10^6 in this context. So, a frequency of 93.7 megahertz is equal to \(93.7 \times 10^6\) hertz.

e. Milligrams (

In this case, the prefix "milli" (symbol: m) indicates a factor of 10^(-3). Therefore, 2 milligrams of active ingredient per cubic centimeter is equal to \(2 \times 10^{-3}\) grams of active ingredient per cubic centimeter.

f. Megabucks (

As in part d, the prefix "mega" (symbol: M) indicates a factor of 10^6. If you won the lottery for "megabucks", it means that you won a prize of \(10^6\) bucks.

Key concepts

Understanding the Metric System

The metric system is an internationally agreed-upon system of measurement based on the power of ten. It is a decimal system of units that is used around the world for scientific and everyday measurements. The simplicity of the metric system lies in its use of prefixes that are attached to basic units of measurement to represent multiples or fractions of those units.

For example, the prefix 'kilo-' signifies a multiple of one thousand, which means that 1 kilogram is 1000 grams. Similarly, 'milli-' corresponds to a thousandth part of a unit, so 1 milligram would be equivalent to 0.001 grams. Learning these prefixes, from 'kilo-' for thousands to 'milli-' for thousandths, enables us to easily switch between different scales of measurement without complex conversions.

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. It consists of two parts: a coefficient and a power of ten. The coefficient is a number greater than or equal to 1 and less than 10, and it is multiplied by 10 raised to an exponent. The exponent indicates how many places the decimal point moves to the right for positive exponents, or to the left for negative exponents.

For instance, the number 93.7, when written as 9.37 × 10¹, is presented in scientific notation. This notation is particularly useful for measurements in the metric system, as the prefixes correspond to powers of ten. For example, 'giga-' equates to 10⁹ and 'mega-' to 10⁶. Understanding how to use scientific notation allows us to work with very large or small numbers efficiently in mathematical and scientific contexts.

Units of Measurement

Units of measurement are the labels we assign to quantities of a particular kind, such as length, mass, volume, or time, allowing us to describe and quantify the world around us. The International System of Units (SI), based on the metric system, defines seven fundamental units from which all other units are derived.

In practice, we often encounter derived units like grams for mass, meters for length, and seconds for time, among others. These standard units provide a common language for science, industry, trade, and everyday use. For instance, when we say a substance has a mass of 2 milligrams per cubic centimeter, we are using the milligram to describe a very small mass, and the cubic centimeter for a specific volume. These standardized units are essential for clear communication, enabling us to replicate experiments, share recipes, or navigate efficiently.