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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

Expert verified
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

01

(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: \(2.3 \mathrm{~pL}\).
02

(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: \(4.7 \mathrm{~µg}\).
03

(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: \(1.85 \mathrm{~pm}\).
04

(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: \(16.7 \mathrm{~Ms}\).
05

(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: \(15.7 \mathrm{~kg}\).
06

(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: \(1.34 \mathrm{~mm}\).
07

(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: \(1.84 \mathrm{~dam}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Suppose you decide to define your own temperature scale with units of \(\mathrm{O}\), using the freezing point \(\left(13{ }^{\circ} \mathrm{C}\right)\) and boiling point \(\left(360^{\circ} \mathrm{C}\right)\) of oleic acid, the main component of olive oil. If you set the freezing point of oleic acid as \(0^{\circ} \mathrm{O}\) and the boiling point as \(100^{\circ} \mathrm{O}\), what is the freezing point of water on this new scale?

The inside dimension of a box that is cubic is \(24.8 \mathrm{~cm}\) on each edge with an uncertainty of \(0.2 \mathrm{~cm}\). What is the volume of the box? What do you estimate to be the uncertainty in the calculated volume?

The total rate at which power used by humans worldwide is approximately \(15 \mathrm{TW}\) (terawatts). The solar flux averaged over the sunlit half of Earth is \(680 \mathrm{~W} / \mathrm{m}^{2}\). (assuming no clouds). The area of Earth's disc as seen from the sun is \(1.28 \times 10^{14} \mathrm{~m}^{2}\). The surface area of Earth is approximately \(197,000,000\) square miles. How much of Earth's surface would we need to cover with solar energy collectors to power the planet for use by all humans? Assume that the solar energy collectors can convert only \(10 \%\) of the available sunlight into useful power.

In the year 2011, an estimated amount of 35 billion tons of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) was emitted worldwide due to fossil fuel combustion and cement production. Express this mass of \(\mathrm{CO}_{2}\) in grams without exponential notation, using an appropriate metric prefix.

(a) To identify a liquid substance, a student determined its density. Using a graduated cylinder, she measured out a \(45-\mathrm{mL}\) sample of the substance. She then measured the mass of the sample, finding that it weighed \(38.5 \mathrm{~g}\). She knew that the substance had to be either isopropyl alcohol (density \(0.785 \mathrm{~g} / \mathrm{mL}\) ) or toluene (density \(0.866 / \mathrm{mL}\) ). What are the calculated density and the probable identity of the substance? (b) An experiment requires \(45.0 \mathrm{~g}\) of ethylene glycol, a liquid whose density is \(1.114 \mathrm{~g} / \mathrm{mL}\). Rather than weigh the sample on a balance, a chemist chooses to dispense the liquid using a graduated cylinder. What volume of the liquid should he use? (c) Is a graduated cylinder such as that shown in Figure \(1.19\) likely to afford the accuracy of measurement needed? (d) A cubic piece of metal measures \(5.00 \mathrm{~cm}\) on each edge. If the metal is nickel, whose density is \(8.90 \mathrm{~g} / \mathrm{cm}^{3}\), what is the mass of the cube?

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