Question

Use prefix multipliers to express each measurement without exponents. a. \(38.8 \times 10^{5} \mathrm{~g}\) b. \(55.2 \times 10^{-10} \mathrm{~s}\) c. \(23.4 \times 10^{11} \mathrm{~m}\) d. \(87.9 \times 10^{-7} \mathrm{~L}\)

Short answer

a. 3.88 Mg, b. 552 ps, c. 234 Gm, d. 879 μL

Step-by-step solution

Understanding Prefix Multipliers

Prefix multipliers are used in the metric system to express large or small numbers in a more manageable form. Each prefix has a corresponding factor of 10. We will associate each measurement with the appropriate prefix based on its exponent.

Converting Grams

For measurement a. (\(38.8 \times 10^{5} \text{ g}\)), the exponent on 10 is 5. This corresponds to the prefix 'mega' (\text{M}), which represents \times 10^{6}. So to express \times 10^{5}, we move the decimal one place to the left and use the prefix 'mega' resulting in 3.88 megagrams (Mg).

Converting Seconds

For measurement b. (\(55.2 \times 10^{-10} \text{ s}\)), the exponent on 10 is -10. This corresponds closely to the prefix 'pico' (\text{p}), which represents \times 10^{-12}. To match the exponent, we move the decimal two places to the right and use 'pico' resulting in 552 picoseconds (ps).

Converting Meters

For measurement c. (\(23.4 \times 10^{11} \text{ m}\)), the exponent is 11. The prefix 'giga' (\text{G}) corresponds to \times 10^{9}. Moving the decimal two places to the left to match the exponent with the prefix, we get 234 gigameters (Gm).

Converting Liters

For measurement d. (\(87.9 \times 10^{-7} \text{ L}\)), the exponent is -7. The prefix 'micro' (\text{μ}) corresponds to \times 10^{-6}. To match the exponent, we move the decimal one place to the right and use 'micro', resulting in 879 microliters (μL).

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It uses powers of ten to simplify the representation of numbers. This form is especially useful in chemistry and physics where extreme values, such as the speed of light or the size of atoms, are common.

For example, the number 6,000,000,000 can be written as \(6 \times 10^{9}\) in scientific notation, where the number 6 is called the coefficient and the exponent 9 indicates how many places the decimal point has been moved to the right. Similarly, 0.0000009 would be \(9 \times 10^{-7}\), where the decimal point is moved 7 places to the left. Scientific notation allows for easier multiplication and division, as you can add and subtract exponents when dealing with powers of ten.

Metric System

The metric system is an internationally adopted system of measurement based on multiples of ten. It is also known as the International System of Units (SI). The key feature of the metric system is its use of prefixes like 'kilo-' (\(10^3\)), 'mega-' (\(10^6\)) and 'micro-' (\(10^{-6}\)) to denote different scales of measurement.

This system simplifies the conversion between units because it is based on powers of ten. For example, converting from meters to kilometers is straightforward because 1 kilometer is exactly 1,000 meters. This coherence allows for a simplified approach to calculation and communication in science and engineering, as everyone uses a common base for measurements.

Unit Conversion

Unit conversion is a multi-step process for converting one unit of measurement into another. In the context of the metric system and scientific notation, unit conversion involves using prefix multipliers. To convert from one metric unit to another, you either multiply or divide by factors of ten, depending on the direction of the conversion.

For instance, to convert 3.88 megagrams (Mg) back to grams (g), you would multiply by \(10^6\) because 'mega' implies a million, or \(10^6\). Conversely, if you want to convert 879 microliters (μL) to liters (L), you divide by \(10^6\), since 'micro' means one-millionth. These conversions are critical for scientists as they often need to express measurements in different units to align with standards or to make the values more comprehensible.