Which one of the following is the value of one nanometer?
10 −7 cm
Understanding Nanometer Value in Centimeters
The question asks for the value of one nanometer expressed in centimeters. To solve this, we need to understand the relationship between different units of length in the metric system.
Unit Conversion Basics: Nanometer to Meter
A nanometer (nm) is a very small unit of length. The prefix 'nano' means \(10^{-9}\). Therefore, one nanometer is equal to \(10^{-9}\) meters.
- 1 nanometer (nm) = \(10^{-9}\) meters (m)
Unit Conversion Basics: Meter to Centimeter
A centimeter (cm) is also a unit of length. The prefix 'centi' means \(10^{-2}\). Therefore, one centimeter is equal to \(10^{-2}\) meters, or equivalently, one meter is equal to 100 centimeters.
- 1 meter (m) = 100 centimeters (cm)
- 1 meter (m) = \(10^2\) centimeters (cm)
Converting Nanometers to Centimeters Step-by-Step
Now we can convert nanometers to centimeters by using the meter as an intermediate unit:
- Start with the value in nanometers: 1 nm
- Convert nanometers to meters: 1 nm = \(10^{-9}\) m
- Convert meters to centimeters: We know 1 m = \(10^2\) cm. So, to convert \(10^{-9}\) meters to centimeters, we multiply by \(10^2\).
The calculation is as follows:
\(1 \text{ nm} = 10^{-9} \text{ m}\)
\(1 \text{ m} = 10^2 \text{ cm}\)
Substitute the second equation into the first:
\(1 \text{ nm} = 10^{-9} \times (10^2 \text{ cm})\)
Using the rule of exponents \(x^a \times x^b = x^{a+b}\):
\(1 \text{ nm} = 10^{-9 + 2} \text{ cm}\)
\(1 \text{ nm} = 10^{-7} \text{ cm}\)
Comparing with Options
Let's compare our calculated value with the given options:
| Option | Value | Comparison |
|---|---|---|
| 1 | \(10^{-7}\) cm | Matches our calculation |
| 2 | \(10^{-6}\) cm | Does not match |
| 3 | \(10^{-4}\) cm | Does not match |
| 4 | \(10^{-3}\) cm | Does not match |
The calculated value of one nanometer is \(10^{-7}\) cm, which matches Option 1.
Revision Table: Unit Conversions
| Unit | Symbol | Equivalent in Meters | Equivalent in Centimeters |
|---|---|---|---|
| Meter | m | 1 m | \(10^2\) cm (100 cm) |
| Centimeter | cm | \(10^{-2}\) m | 1 cm |
| Nanometer | nm | \(10^{-9}\) m | \(10^{-7}\) cm |
| Micrometer (Micron) | μm | \(10^{-6}\) m | \(10^{-4}\) cm |
| Millimeter | mm | \(10^{-3}\) m | \(10^{-1}\) cm |
Additional Information: Metric Prefixes
The metric system uses prefixes to denote multiples or submultiples of base units. Understanding these prefixes is crucial for unit conversions. Here are some common prefixes related to length:
- Giga (G): \(10^9\)
- Mega (M): \(10^6\)
- Kilo (k): \(10^3\)
- Hecto (h): \(10^2\)
- Deca (da): \(10^1\)
- Deci (d): \(10^{-1}\)
- Centi (c): \(10^{-2}\)
- Milli (m): \(10^{-3}\)
- Micro (μ): \(10^{-6}\)
- Nano (n): \(10^{-9}\)
- Pico (p): \(10^{-12}\)
When converting between units, you can move the decimal point or multiply/divide by powers of 10 based on the difference in the exponents of the prefixes relative to the base unit (like the meter). For example, to go from meters (\(10^0\)) to centimeters (\(10^{-2}\)), you multiply by \(10^{0 - (-2)} = 10^2\). To go from meters (\(10^0\)) to nanometers (\(10^{-9}\)), you multiply by \(10^{0 - (-9)} = 10^9\), meaning 1 meter is \(10^9\) nanometers. Conversely, 1 nanometer is \(10^{-9}\) meters.
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