Question

Use scientific notation to express each quantity with only base units (no prefix multipliers). MISSED THIS? Read Section 1.6; Watch KCV 1.6 a. \(4.5 \mathrm{~ns}\) b. 18 fs c. \(128 \mathrm{pm}\) d. \(35 \mu \mathrm{m}\)

Short answer

a. 4.5 x 10^-9 s; b. 18 x 10^-15 s; c. 128 x 10^-12 m; d. 35 x 10^-6 m.

Step-by-step solution

Convert Nanoseconds to Seconds

First thing we need to do is to convert the time from nanoseconds to seconds. Since 1 nanosecond (ns) is equal to 1x10^-9 seconds, we write it in scientific notation: 4.5 ns = 4.5 x 10^-9 s.

Convert Femtoseconds to Seconds

To convert femtoseconds to seconds, use the conversion where 1 femtosecond (fs) equals 1x10^-15 seconds. The conversion is as follows: 18 fs = 18 x 10^-15 s.

Convert Picometers to Meters

To change the unit from picometers to meters, note that 1 picometer (pm) equals 1x10^-12 meters. Hence, 128 pm = 128 x 10^-12 m.

Convert Micrometers to Meters

Lastly, to convert from micrometers to meters, we use the fact that 1 micrometer (µm) equals 1x10^-6 meters. Therefore, 35 µm = 35 x 10^-6 m.

Key concepts

Unit Conversion

Unit conversion is a fundamental skill in scientific studies as it allows us to express measurements in different but equivalent units. This makes comparisons and calculations more straightforward, especially when working within complex systems or converting between systems of measurement (like from the metric to the imperial system).

Using scientific notation in unit conversion is particularly useful because it simplifies working with very large or very small numbers. Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10, making them easier to manipulate and understand.

For example, in the solution for the given exercise, nanoseconds were converted to seconds using scientific notation: 4.5 ns becomes \(4.5 \times 10^{-9} \text{s}\). This conversion is essential for ensuring that measurements are in the base units, which are standard units of measurement within a particular system, such as the International System of Units (SI) for scientific and technical work.

Base Units

Base units form the foundation of any system of measurement. In the International System of Units (SI), there are seven base units, which encompass the meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for the amount of substance, and candela (cd) for luminous intensity.

All other units are derived from these base units, allowing for a wide range of measurements. For instance, when we look at the conversion of 35 micrometers to meters, the meter is the base unit for length. By converting \(35 \mu m\) to meters (\(35 \times 10^{-6} m\)), we express the length in base units, which is critical for uniformity and aiding in scientific communication and collaboration.

Prefix Multipliers

Prefix multipliers are shorthand representations that make handling and communicating very large or very small numbers more manageable. They denote a specific power of ten, allowing quick conversions by simply shifting the decimal point up or down as per the multiplier's power of ten.

Common prefix multipliers include 'nano' (\(10^{-9}\)), 'micro' (\(10^{-6}\)), 'milli' (\(10^{-3}\)), 'centi' (\(10^{-2}\)), 'kilo' (\(10^{3}\)), and 'mega' (\(10^{6}\)). When converting units, such as in the exercise 18 fs (femtoseconds) being converted to seconds, the 'femto' prefix represents \(10^{-15}\). Therefore, we can express this as \(18 \times 10^{-15} s\).

Understanding prefix multipliers is vital for working with units of measurement since it allows scientists and students to quickly adapt and calculate different magnitudes of measurements within and across different unit systems.