Question

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(2.3 \times 10^{-10} \mathrm{L}\) ,(b) \(4.7 \times 10^{-6} \mathrm{g},\) (c) \(1.85 \times 10^{-12} \mathrm{m},\) (d) \(16.7 \times 10^{6} \mathrm{s}\) (e) \(15.7 \times 10^{3} \mathrm{g},(\mathrm{f}) 1.34 \times 10^{-3} \mathrm{m},(\mathrm{g}) 1.84 \times 10^{2} \mathrm{cm}\)

Short answer

(a) \(230 \,\mathrm{nL}\), (b) \(4.7 \,\mathrm{µg}\), (c) \(1.85 \,\mathrm{pm}\), (d) \(16.7 \,\mathrm{Ms}\), (e) \(15.7 \,\mathrm{kg}\), (f) \(1.34 \,\mathrm{mm}\), (g) \(1.84 \,\mathrm{hm}\)

Step-by-step solution

a)

Convert \(2.3 \times 10^{-10} \mathrm{L}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(-10\), we will use nano (n) prefix: \[2.3 \times 10^{-10} \mathrm{L} = 230 \,\mathrm{nL}\]

b)

Convert \(4.7 \times 10^{-6} \mathrm{g}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(-6\), we will use micro (µ) prefix: \[4.7 \times 10^{-6} \mathrm{g} = 4.7 \,\mathrm{µg}\]

c)

Convert \(1.85 \times 10^{-12} \mathrm{m}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(-12\), we will use pico (p) prefix: \[1.85 \times 10^{-12} \mathrm{m} = 1.85 \,\mathrm{pm}\]

d)

Convert \(16.7 \times 10^{6} \mathrm{s}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(6\), we will use Mega (M) prefix: \[16.7 \times 10^{6} \mathrm{s} =16.7 \,\mathrm{Ms}\]

e)

Convert \(15.7 \times 10^{3} \mathrm{g}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(3\), we will use kilo (k) prefix: \[15.7 \times 10^{3} \mathrm{g} = 15.7 \,\mathrm{kg}\]

f)

Convert \(1.34 \times 10^{-3} \mathrm{m}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(-3\), we will use milli (m) prefix: \[1.34 \times 10^{-3} \mathrm{m} = 1.34 \,\mathrm{mm}\]

g)

Convert \(1.84 \times 10^{2} \mathrm{cm}\) to an equivalent value using appropriate metric prefix. Since the exponent is \(2\), we will use hecto (h) prefix. Note that 1 hectometer (hm) = 10000 centimeters (cm): \[1.84 \times 10^{2} \mathrm{cm} = 1.84 \,\mathrm{hm}\]

Key concepts

Scientific Notation

When dealing with very large or very small numbers, scientists and engineers use scientific notation to make these numbers more manageable. Scientific notation expresses numbers as a product of two parts: a coefficient and a power of ten. For example, the number 5000 can be written in scientific notation as \(5 \times 10^3\). This format is particularly helpful in simplifying complex calculations, as it allows for straightforward multiplication and division.

Numbers smaller than one are also easily represented in scientific notation. For instance, 0.00012 becomes \(1.2 \times 10^{-4}\). Each step in the exponent represents a shift of the decimal point by one place. Using scientific notation aligns with our decimal system, making it a universal tool for scientists around the world in all fields ranging from physics to economics.

Unit Conversion

Unit conversion is a critical technique in many scientific calculations, allowing for comparison and computation across different measurement systems. Unit conversion involves the use of multiplication or division to convert one unit to another while maintaining the same quantity. This is often achieved by utilising conversion factors that are based on relationships between units. For example, knowing that 1 kilometer is equal to 1000 meters, one can convert 5 kilometers to meters by multiplying 5 by 1000, yielding 5000 meters.

To ensure accuracy, the unit you are converting to should cancel out the unit you are converting from. When working with scientific notation, this process also means adjusting the exponent as needed to reflect the new unit's scale.

Metric System

The metric system is an internationally agreed-upon decimal system of measurement. It is used almost universally around the world, which simplifies international communication and trade. One of the system's key features is its use of metric prefixes, which indicate a multiple of the base unit. These prefixes serve as shorthand to express large or small quantities.

Common metric prefixes include:

• Kilo (k): Represents 1,000 units of the base
• Mega (M): Equals 1,000,000 units of the base
• Micro (µ): Means one millionth of the base
• Nano (n): Signifies one billionth of the base
• Pico (p): Indicates one trillionth of the base

Instead of writing out the whole number, such as 0.000001 meters, we can use the metric prefix to simplify it to 1 micrometer (1 µm). This simplicity is one of the reasons why the metric system is preferred in science and engineering.