Question

Use the prefix multipliers to express each measurement without exponents. MISSED THIS? Read Section 1.6; Watch \(\mathrm{KCV} 1.6\) a. \(1.2 \times 10^{-9} \mathrm{~m}\) b. \(22 \times 10^{-15} \mathrm{~s}\) c. \(1.5 \times 10^{9} \mathrm{~g}\) d. \(3.5 \times 10^{6} \mathrm{~L}\)

Short answer

a. 1.2 nanometers (nm) b. 22 femtoseconds (fs) c. 1.5 gigagrams (Gg) d. 3.5 megaliters (ML)

Step-by-step solution

Understanding Prefix Multipliers

Prefix multipliers are used in the metric system to represent different powers of 10. They replace the scientific notation to express the magnitude of a measurement in a more readable way. For example, 'kilo' means a thousand times (10^3), 'milli' means a thousandth of (10^-3), and so on.

Convert 1.2 x 10^-9 m to Nanometers

The prefix for 10^-9 is 'nano' (n). Therefore, to convert meters to nanometers, multiply the number by 10^9. In this case, 1.2 x 10^-9 meters is equal to 1.2 nanometers.

Convert 22 x 10^-15 s to Femtoseconds

The prefix for 10^-15 is 'femto' (f). To convert seconds to femtoseconds, multiply the number by 10^15. Hence, 22 x 10^-15 seconds is 22 femtoseconds.

Convert 1.5 x 10^9 g to Gigagrams

The prefix for 10^9 is 'giga' (G). To convert grams to gigagrams, divide the number by 10^9. Therefore, 1.5 x 10^9 grams equals 1.5 gigagrams.

Convert 3.5 x 10^6 L to Megaliters

The prefix for 10^6 is 'mega' (M). To convert liters to megaliters, divide the number by 10^6. So, 3.5 x 10^6 liters is the same as 3.5 megaliters.

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or small to be conveniently written in decimal form. It is widely used in science, engineering, and mathematics to simplify the handling of big numbers, like the distance between stars, or small ones, like the mass of an atom.

At its core, scientific notation consists of two parts: a coefficient and a power of ten. The coefficient is a number between 1 and 10, and the exponent indicates how many times the coefficient should be multiplied by ten. For instance, the number 600 can be written as 6 x 10² in scientific notation because 6 multiplied by 10 squared (10 times 10) equals 600.

In the context of our exercise, the number 1.2 x 10^-9 m indicates a very small measurement that is expedient to write using scientific notation, rather than writing out many zeros which would be required in standard decimal form. Learning to interpret and convert these notations is fundamental in handling data efficiently in science and engineering.

Metric System

The metric system is an internationally adopted decimal system of measurement. It is based on multiples of ten and includes a set of units and prefixes that are used to express the scale of measurements. This system is standardized so that it is easily understood and globally consistent.

Prefix multipliers, such as kilo, centi, and milli, are attached to basic metric units like meters, grams, and liters to indicate multiples or fractions of these units. For example, the 'kilo-' prefix signifies 1,000 units, so 1 kilometer (km) is 1,000 meters (m).

Importance of the Metric System

To ensure clear communication and minimize errors, using the metric system is essential in science and technology. Because it is decimal, it makes calculations simpler and more intuitive. The metric system also enables us to quickly convert between different scales of measurement by simply moving the decimal point, rather than performing complex multiplication or division.

Unit Conversion

Unit conversion is the process of changing from one unit to another within the same measurement system, often involving multiplicative conversion factors. In our exercise, unit conversion requires knowledge of the prefixes and an understanding of the power of tens they represent.

Knowing that 'nano-' means 10^-9, 'femto-' means 10^-15, 'giga-' means 10⁹, and 'mega-' means 10⁶ allows us to convert metrics into more tangible quantities. Essentially, this is a two-step process: first, identify the correct prefix for the exponent, and second, perform the appropriate multiplication or division.

Applying Conversion

When approaching a problem like converting 1.5 x 10⁹ grams to Gigagrams, we identify that giga represents a billion (10⁹), and then divide the value by a billion to find the equivalent in Gigagrams. Similarly, converting 3.5 x 10⁶ liters to Megaliters involves recognizing that 'mega' represents a million (10⁶), and dividing by a million. Mastering unit conversion is a valuable skill, ensuring accuracy and precision in scientific measurements and calculations.