Question

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(6.35 \times 10^{-2} \mathrm{~L}\), (b) \(6.5 \times 10^{-6} \mathrm{~s}\), (c) \(9.5 \times 10^{-4} \mathrm{~m}\), (d) \(4.23 \times 10^{-9} \mathrm{~m}^{3}\), (e) \(12.5 \times 10^{-8} \mathrm{~kg}\), (f) \(3.5 \times 10^{-10} \mathrm{~g}\) (g) \(6.54 \times 10^{9} \mathrm{fs}\).

Short answer

(a) 6.35 cL (b) 6.5 μs (c) 0.95 mm (d) 4.23 nm³ (e) 125 μg (f) 0.35 ng (g) 6.54 ns

Step-by-step solution

(a) Convert 6.35 x 10^{-2} L to an equivalent form with a metric prefix

To convert this measurement, we look for the appropriate metric prefix for the exponent, -2. We can see that this is the centi (c) prefix, which corresponds to 10^{-2}. So we rewrite the measurement as: 6.35 x 10^{-2} L = 6.35 cL

(b) Convert 6.5 x 10^{-6} s to an equivalent form with a metric prefix

In this case, the appropriate metric prefix for the exponent, -6, is micro (μ) which corresponds to 10^{-6}. So we rewrite this measurement as: 6.5 x 10^{-6} s = 6.5 μs

(c) Convert 9.5 x 10^{-4} m to an equivalent form with a metric prefix

Here, the appropriate metric prefix for the exponent, -4, is milli (m) which corresponds to 10^{-3}. So we rewrite this measurement as: 9.5 x 10^{-4} m = 0.95 mm

(d) Convert 4.23 x 10^{-9} m^3 to an equivalent form with a metric prefix

For this measurement, the appropriate metric prefix for the exponent -9 is nano (n) which corresponds to 10^{-9}. So we rewrite the measurement as: 4.23 x 10^{-9} m^3 = 4.23 nm^3

(e) Convert 12.5 x 10^{-8} kg to an equivalent form with a metric prefix

For this measurement, the appropriate metric prefix for the exponent, -8, is between micro (μ) and nano (n), which corresponds to 10^{-6} and 10^{-9}, respectively. We can rewrite the measurement as: 12.5 x 10^{-8} kg = 125 x 10^{-9} kg = 125 μg (microgram)

(f) Convert 3.5 x 10^{-10} g to an equivalent form with a metric prefix

Here, the appropriate metric prefix for the exponent, -10, is between nano (n) and pico (p), which corresponds to 10^{-9} and 10^{-12}, respectively. We can rewrite the measurement as: 3.5 x 10^{-10} g = 0.35 x 10^{-9} g = 0.35 ng (nanogram)

(g) Convert 6.54 x 10^9 fs to an equivalent form with a metric prefix

For this measurement, the appropriate metric prefix for the exponent, 9, is giga (G) which corresponds to 10^9. However, in this case, the measurement is in femtoseconds (fs) which is already a metric prefix. We can rewrite the measurement as: 6.54 x 10^9 fs = 6.54 ns (nanosecond)

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It's widely used in sciences for ease of calculation and simplification of data readings.
For instance, a number like 0.000001 can be expressed in scientific notation as \(1 \times 10^{-6}\). The format consists of a coefficient (a digit or two) and an exponent of ten.

• The coefficient is typically a number between 1.0 and 10.0
• The exponent indicates how many times the coefficient should be multiplied or divided by ten

Scientific notation simplifies complex numbers, making it easier to read, write, and compute especially in scientific calculations where measurements often range across several orders of magnitude.

Metric System

The metric system is a universal system of measurements used worldwide for scientific and everyday measurements. It is based on powers of ten, which makes conversions straightforward within the system. Each unit of measurement can be easily converted into another by the use of prefixes. These prefixes indicate the scale or magnitude of the measurement.
The main metric prefixes include:

• Kilo (k) for \(10^3\)
• Hecto (h) for \(10^2\)
• Deca (da) for \(10^1\)
• Deci (d) for \(10^{-1}\)
• Centi (c) for \(10^{-2}\)
• Milli (m) for \(10^{-3}\)
• Micro (\(\mu\)) for \(10^{-6}\)
• Nano (n) for \(10^{-9}\)
• Pico (p) for \(10^{-12}\)

Using these prefixes helps to express measurements without using long series of decimal places, and it allows for easy conversion between different scales.

Measurement Conversion

Measurement conversion within the metric system is straightforward due to its base-10 nature. Converting between different metric units simply involves shifting the decimal point or adjusting the exponent in scientific notation.
This consistency makes it simple to switch between units, for example, from millimeters to meters, as both are defined by powers of ten.

• When converting to a larger unit, you shift the decimal to the left.
• When converting to a smaller unit, you shift the decimal to the right.

This principle is useful in practical applications, such as scientific research and daily life calculations where precise conversions are necessary. Mastery of these concepts allows for quick and accurate conversions that streamline both learning and application of the metric system.