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) \(15.7 \times 10^{3} \mathrm{~g}\), (f) \(1.34 \times 10^{-3} \mathrm{~m}\), (g) \(1.84 \times 10^{2} \mathrm{~cm}\)

Short answer

The short answer is: (a) \(2.3 \mathrm{~pL}\), (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) \(1.84 \mathrm{~dam}\).

Step-by-step solution

(a) Rewrite 2.3 x 10^{-10} L using a metric prefix

To do this, we will identify the appropriate metric prefix for the given exponent, which in this case is -10. The metric prefix for 10^{-10} is pico (p). So we can rewrite the given measurement as: \(2.3 \mathrm{~pL}\).

(b) Rewrite 4.7 x 10^{-6} g using a metric prefix

The appropriate metric prefix for the exponent -6 is micro (µ). Therefore, the given measurement can be rewritten as: \(4.7 \mathrm{~µg}\).

(c) Rewrite 1.85 x 10^{-12} m using a metric prefix

The metric prefix for the exponent -12 is pico (p). So we rewrite the given measurement as: \(1.85 \mathrm{~pm}\).

(d) Rewrite 16.7 x 10^6 s using a metric prefix

The metric prefix corresponding to the exponent 6 is mega (M). Therefore, the given measurement can be rewritten as: \(16.7 \mathrm{~Ms}\).

(e) Rewrite 15.7 x 10^3 g using a metric prefix

In this case, the exponent is 3, which corresponds to the metric prefix kilo (k). We rewrite the given measurement as: \(15.7 \mathrm{~kg}\).

(f) Rewrite 1.34 x 10^{-3} m using a metric prefix

The metric prefix for the exponent -3 is milli (m). Therefore, we rewrite the given measurement as: \(1.34 \mathrm{~mm}\).

(g) Rewrite 1.84 x 10^2 cm using a metric prefix

The exponent 2 in this case corresponds to the metric prefix hecto (h). However, hecto is typically not used with centimeters. So first, let's convert centimeters to meters. We know that 1 cm = 0.01 m, so we have: \(1.84 \times 10^2 \mathrm{~cm} = (1.84 \times 10^2) \times (0.01 \mathrm{~m})\) Multiply the numbers: \(1.84 \times 10^2 \times 0.01 = 1.84 \times 10\) Now, we rewrite the given measurement with the corresponding metric prefix for the exponent 1, which is deka (da). So we have: \(1.84 \mathrm{~dam}\).

Key concepts

Pico

The term "pico" is a metric prefix that denotes a factor of \(10^{-12}\). It is often used to describe very small quantities, sometimes in the realms of physics and biology.
When we work with measurements in the order of tera, giga, or even mega, the sizes can range massively, hence why there is a need for prefixes like pico to effectively communicate small scales.

For example, in chemistry, you might come across picometers (pm), which are used to measure atomic lengths, or picofarads in electronics to describe capacitors.
It is very important to use these prefixes correctly as they accurately communicate the size and scale of the measurement.
Back to our exercise, when dealing with numbers such as \(1.85 \times 10^{-12}\) m, the measurement becomes 1.85 pm with the pico prefix.

Micro

"Micro" is a metric prefix representing \(10^{-6}\) or one millionth of a unit.
It's commonly seen in many scientific fields, particularly in biology with terms like microliters (µL) and micrograms (µg), as well as in physics and chemistry.
Due to its widespread application in various sciences, understanding the micro prefix becomes crucial for interpreting data correctly.

In our context, when you see something like \(4.7 \times 10^{-6}\) grams, this is equivalent to 4.7 micrograms (\(\mathrm{µg}\)).
This usage simplifies communicating and visualizing massively small measurements. Understanding and recognizing the micro prefix helps with reading and calculating these very small, essential quantities.

Milli

The "milli" prefix stands for \(10^{-3}\) or one thousandth of a unit.
It appears frequently in everyday measures like milliliters (mL) and millimeters (mm), providing a way to talk about small quantities in practical settings.

Using the milli prefix allows for easy comprehension and swift communication.
In our exercise example, \(1.34 \times 10^{-3}\) meters translates to 1.34 millimeters (\(\mathrm{mm}\)). This simplification shows how these prefixes serve not only the scientific community but also every day measurements.

Mega

The metric prefix "mega" signifies \(10^{6}\), or a million times a base unit.
This prefix is usually used when dealing with large quantities, offering clarity and simplification in representation. We often use mega in terms like megabytes (MB) to describe data size, or megawatts (MW) to express power.

When looking at the original exercise, \(16.7 \times 10^{6}\) seconds can be rewritten as 16.7 megaseconds (\(\mathrm{Ms}\)).
This adjustment reflects a shift from a numerical figure to a more intuitive and comprehensible term, making the measurement immediately understandable.

Kilo

The prefix "kilo" denotes \(10^{3}\), which is a thousand times the base unit.
You likely hear this prefix in everyday contexts, like kilograms (kg) and kilometers (km). This unit of measure is highly prevalent, as it provides a straightforward understanding of quantities that are larger.

In the exercise provided, \(15.7 \times 10^{3}\) grams translates to 15.7 kilograms (\(\mathrm{kg}\)).
Using "kilo" allows individuals to easily understand the scale of the measurement without getting tangled in zeros. Ultimately, the metric system, with its prefixes like kilo, facilitates universal and manageable communication in science, industry, and everyday life.