Question

For each of the following descriptions, identify the power of 10 being indicated by the prefix in the measurement. a. The sign on the interstate highway says to tune my AM radio to 540 kilohertz for traffic information. b. My new digital camera has a 2 -gigabyte flash memory card. c. The shirt I bought for my dad on my European vacation shows the sleeve length in centimeters. d. My brother's camcorder records on 8 -millimeter tape cassettes.

Short answer

a. The prefix "kilo" corresponds to the power of 10 equal to 3. b. The prefix "giga" corresponds to the power of 10 equal to 9. c. The prefix "centi" corresponds to the power of 10 equal to -2. d. The prefix "milli" corresponds to the power of 10 equal to -3.

Step-by-step solution

a. Identify the prefix in 540 kilohertz

In this situation, the prefix is "kilo" which corresponds to a frequency of 540 kilohertz (kHz). "Kilo" means a thousand, so we need to find the power of 10 that represents a thousand.

a. Determine the power of 10 for "kilo"

As "kilo" means a thousand, we will look for the power of 10 that gives us 1000. \(10^3 = 1000\), therefore, the prefix "kilo" corresponds to the power of 10 equal to 3.

b. Identify the prefix in 2-gigabyte flash memory card

In this situation, the prefix is "giga" which corresponds to a memory capacity of 2 gigabytes (GB). "Giga" means a billion, so we need to find the power of 10 that represents a billion.

b. Determine the power of 10 for "giga"

As "giga" means a billion, we will look for the power of 10 that gives us 1,000,000,000. \(10^9 = 1,000,000,000\), therefore, the prefix "giga" corresponds to the power of 10 equal to 9.

c. Identify the prefix in the sleeve length in centimeters

In this situation, the prefix is "centi" which corresponds to the sleeve length in centimeters. "Centi" means one hundredth (1/100), so we need to find the power of 10 that represents one hundredth.

c. Determine the power of 10 for "centi"

As "centi" means one hundredth, we will look for the power of 10 that gives 1/100. From this, we get \(10^{-2} = 1/100\). Therefore, the prefix "centi" corresponds to the power of 10 equal to -2.

d. Identify the prefix in 8-millimeter tape cassettes

In this situation, the prefix is "milli" which corresponds to the tape length of 8 millimeters (mm). "Milli" means one thousandth (1/1000), so we need to find the power of 10 that represents one thousandth.

d. Determine the power of 10 for "milli"

As "milli" means one thousandth, we will look for the power of 10 that gives 1/1000. From this, we get \(10^{-3} = 1/1000\). Therefore, the prefix "milli" corresponds to the power of 10 equal to -3.

Key concepts

Powers of 10

Understanding powers of 10 can make working with the metric system a lot easier. Powers of 10 serve as a shorthand way of expressing numbers using a base of 10. When we talk about powers of 10, it's simply writing numbers as an exponent of 10. For example:

• The power of 10 for "kilo" is 3, which means 10 to the power of 3 gives us 1,000.
• Similarly, "giga" means a billion, represented as a power of 10 to the 9th power, i.e., 1,000,000,000.
• On the smaller side, "centi" and "milli" stand for one hundredth and one thousandth respectively. So their powers of 10 are -2 and -3.

This system is efficient because it allows us to easily convert between large or small numbers without confusion over the number of zeros involved. Just remember: a positive exponent means you're multiplying by 10, while a negative exponent indicates division.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. It is particularly useful in science and engineering to easily communicate measurements and calculations. Here's the basic idea:

• A number like 5,400 can be written as \( 5.4 \times 10^3 \). Here, 5.4 is the coefficient, and the exponent shows how many times 10 is multiplied by itself.
• For smaller numbers, like 0.01, you would write it as \( 1 \times 10^{-2} \). This indicates division, with the exponent telling you how many times to divide by 10.

By using scientific notation, students can grasp and work with large or small values without getting overwhelmed by digits. It reduces complex numbers to a simpler form, reflecting both the significant digits and the scale in a format that's easy to read and communicate.

Measurement Units

Measurement units are essential in understanding and applying the metric system effectively. These units help standardize what we measure so everyone can easily communicate dimensions, capacities, and other quantities. Let's delve into some specifics:

• In the metric system, every unit is based on powers of 10. For instance, 1 kilometer is 1,000 meters, making 'kilo' a simple indicator of multiplying by 1,000.
• Similarly, a centimeter is one hundredth of a meter, using the prefix 'centi' to denote division by 100.
• For more tiny measurements, like millimeters, the 'milli' prefix indicates one thousandth of a meter.

This system makes conversions straightforward; just adjust the decimal point according to the power of 10 indicated by the prefix. This consistent usage of decimal-based prefixes ensures a unified approach, reducing complexity when dealing with different measurement scales.