Question

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(7.29 \times 10^{6} \mathrm{~g}\) (b) \(6.1 \times 10^{-10} \mathrm{~m}\) (c) \(1.828 \times 10^{-3} \mathrm{~s}\) (d) \(3.523 \times 10^{9} \mathrm{~m}^{3}\) (g) \(3.552 \times 10^{12} \mathrm{~L}\) (e) \(9.62 \times 10^{2} \mathrm{~m} / \mathrm{s}(\mathbf{f}) 8.923 \times 10^{-12} \mathrm{~kg}\)

Short answer

(a) \(7.29 \mathrm{~Mg}\) (b) \(610.0 \mathrm{~nm}\) (c) \(1.828 \mathrm{~ms}\) (d) \(3.523 \mathrm{~Gm}^{3}\) (g) \(3.552 \mathrm{~TL}\) (e) \(0.962 \mathrm{~km/s}\) (f) \(8.923 \mathrm{~pg}\)

Step-by-step solution

(a) Convert g to appropriate metric prefix

To convert \(7.29 \times 10^{6} \mathrm{~g}\), we will use the mega (M) prefix which corresponds to 10^6. Divide by 10^6 and add the prefix: \[ 7.29 \times 10^{6} \mathrm{~g} = 7.29 \mathrm{~Mg} \]

(b) Convert m to appropriate metric prefix

To convert \(6.1 \times 10^{-10} \mathrm{~m}\), we will use the nano (n) prefix which corresponds to 10^-9. Multiply by 10^9 to remove the -10 exponent and add the prefix: \[ 6.1 \times 10^{-10} \mathrm{~m} = 610.0 \mathrm{~nm} \]

(c) Convert s to appropriate metric prefix

To convert \(1.828 \times 10^{-3} \mathrm{~s}\), we will use the milli (m) prefix which corresponds to 10^-3. Divide by 10^-3 and add the prefix: \[ 1.828 \times 10^{-3} \mathrm{~s} = 1.828 \mathrm{~ms} \]

(d) Convert \(m^3\) to appropriate metric prefix

To convert \(3.523 \times 10^{9} \mathrm{~m}^{3}\), we will use the giga (G) prefix which corresponds to 10^9. Divide by 10^9 and add the prefix: \[ 3.523 \times 10^{9} \mathrm{~m}^{3} = 3.523 \mathrm{~Gm}^{3} \]

(g) Convert L to appropriate metric prefix

To convert \(3.552 \times 10^{12} \mathrm{~L}\), we will use the tera (T) prefix which corresponds to 10^12. Divide by 10^12 and add the prefix: \[ 3.552 \times 10^{12} \mathrm{~L} = 3.552 \mathrm{~TL} \]

(e) Convert \(\mathrm{m/s}\) to appropriate metric prefix

To convert \(9.62 \times 10^{2} \mathrm{~m/s}\), we will use the kilo (k) prefix which corresponds to 10^3. Divide by 10^3 with adjustment to maintain the same value and add the prefix: \[ 9.62 \times 10^{2} \mathrm{~m/s} = 0.962 \mathrm{~km/s} \]

(f) Convert kg to appropriate metric prefix

To convert \(8.923 \times 10^{-12} \mathrm{~kg}\), we will use the pico (p) prefix which corresponds to 10^-12. Divide by 10^-12 and add the prefix: \[ 8.923 \times 10^{-12} \mathrm{~kg} = 8.923 \mathrm{~pg} \]

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 is used to simplify calculations and representation, particularly in science and engineering. A number in scientific notation is typically written as \( a \times 10^n \), where:\

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• \( a \) is a number, known as the coefficient, that is generally between 1 and 10.
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• \( n \) is an integer, which represents the number of places the decimal point is moved.
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\By shifting the decimal point using powers of ten, numbers are compacted for easier handling. For example, \(7.29 \times 10^6\) could be simplified to 7,290,000 without losing its significant figures. \(6.1 \times 10^{-10}\) remains much more manageable in this form rather than being expanded into a cumbersome example like 0.00000000061. Understanding scientific notation helps students in seamlessly transitioning to using metric prefixes, a related concept which we will explore in the next section.

Unit Conversion

Unit conversion involves changing a measurement from one unit to another, using some form of calculation that maintains the original value's magnitude. This process relies heavily on conversion factors, which are specific rates or ratios that match two different units that measure the same quantity. It is essential to be familiar with the metric system's prefixes for effective conversion, as often seen in scientific and engineering fields.\
\One example in converting units incorporates using metric prefixes. Suppose we are converting \(6.1 \times 10^{-10}\) meters to nanometers. Knowing that one nanometer equals \(10^{-9}\) meters allows us to derive the conversion factor. We multiply \(6.1 \times 10^{-10}\) by \(10^9\) to cancel out \(10^{-9}\), achieving a result of 610 nanometers.\
\Consistently understanding unit conversions allows students to switch between different units both quickly and accurately, such as going from grams to megagrams or meters to kilometers.

Measurement Units

Measurement units are standard quantities used to quantify physical properties. The International System of Units (SI) is the most commonly used measurement system and is based on the metric system. In this system, units are defined for different types of measurements, such as distance (meters), mass (kilograms), and time (seconds).\
\Units are often modified with prefixes to represent larger or smaller magnitudes of a measurement. For instance, the prefix 'milli-' indicates \(10^{-3}\), thus 1 millisecond (1 ms) is equal to \(1.828 \times 10^{-3}\) seconds. Similarly, larger quantities are expressed in terms like kilometers or megagrams for efficient representation.\
\Choosing the correct measurement unit and its appropriate prefix helps in not only simplifying numbers but also making them more understandable, as seen in scientific communication and various technical fields. Understanding these units is critical for interpretting data and performing calculations accurately across different scientific domains.