Question

Perform the following metric-metric conversions: (a) \(6.50 \mathrm{Tm}\) to \(\mathrm{Mm}\) (b) \(650 \mathrm{Gg}\) to \(\mathrm{kg}\) (c) \(0.650 \mathrm{cL}\) to \(\mathrm{dL}\) (d) 0.000650 ns to ps

Short answer

(a) 6,500,000 Mm, (b) 650,000,000 kg, (c) 0.065 dL, (d) 0.65 ps.

Step-by-step solution

Understand Metric Prefixes

Familiarize yourself with the metric prefixes involved in the conversions. For (a), tera (T) is \(10^{12}\) and mega (M) is \(10^{6}\). For (b), giga (G) is \(10^{9}\) and kilo (k) is \(10^{3}\). For (c), centi (c) is \(10^{-2}\) and deci (d) is \(10^{-1}\). For (d), nano (n) is \(10^{-9}\) and pico (p) is \(10^{-12}\).

Convert Tm to Mm

To convert from tera (\(10^{12}\)) to mega (\(10^{6}\)), realize that \(1 \, ext{Tm} = 10^{12}\, ext{m} = 10^{6}\, ext{Mm}\). Therefore, for \(6.50 \, ext{Tm}\), multiply by \(10^6\) to convert to Mm: \[6.50 \, ext{Tm} = 6.50 imes 10^{6} \, ext{Mm} = 6,500,000 \, ext{Mm}.\]

Convert Gg to kg

To convert from giga (\(10^{9}\)) to kilo (\(10^{3}\)), use the relationship \(1 \, ext{Gg} = 10^{9}\, ext{g} = 10^{6}\, ext{kg}\). Thus, for \(650 \, ext{Gg}\), multiply by \(10^{6}\) to get kilograms: \[650 \, ext{Gg} = 650 imes 10^{6} \, ext{kg} = 650,000,000 \, ext{kg}.\]

Convert cL to dL

To convert from centiliters (\(10^{-2}\)) to deciliters (\(10^{-1}\)), note that a greater magnitude in exponent indicates a smaller actual number. Thus, move \(0.650\) from centiliters to deciliters by dividing by 10:\[0.650 \, ext{cL} = 0.065 \, ext{dL}.\]

Convert ns to ps

To convert from nano (\(10^{-9}\)) to pico (\(10^{-12}\)), use the relationship \(1 \, ext{ns} = 10^{9} \, ext{ps}\). Knowing this, multiply \(0.000650 \, ext{ns}\) by \(10^{3}\) (to account for the difference \(10^{9} - 10^{12} = 10^{3}\)):\[0.000650 \, ext{ns} = 0.000650 imes 10^{3} \, ext{ps} = 0.65 \, ext{ps}.\]

Key concepts

Metric Prefixes

Metric prefixes are essential when understanding metric conversions. They represent powers of ten, making it easier to work with very large or very small quantities.
Each prefix corresponds to a specific power of ten. For example, "tera" represents \(10^{12}\), "giga" signifies \(10^{9}\), "mega" means \(10^{6}\), "kilo" is \(10^{3}\), "centi" indicates \(10^{-2}\), "deci" is \(10^{-1}\), "nano" is \(10^{-9}\), and "pico" signifies \(10^{-12}\).
Understanding these prefixes is crucial because they tell you how many times to multiply or divide a base unit. This standard set of prefixes helps in understanding scientific data and enables universal communication across the scientific community.

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. To convert units, one must use the appropriate conversion factor, often derived from metric prefixes. For example:

• Converting from terameters (Tm) to megameters (Mm) involves recognizing that 1 Tm equals \(10^{6}\) Mm, leading to multiplying by \(10^6\).
• Similarly, converting from gigagrams (Gg) to kilograms (kg) uses the factor that 1 Gg equals \(10^6\) kg.
• From centiliters (cL) to deciliters (dL), you divide by 10 because \(1\, \text{cL} = 0.1\, \text{dL}\).
• For nanoseconds (ns) to picoseconds (ps), multiply by \(10^3\) because \(1\, \text{ns} = 1000\, \text{ps}\).

Breaking down the conversion steps helps minimize errors and ensures accuracy in calculations.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is particularly useful in expressing results of metric conversions.
Using scientific notation, numbers are written as a product of a coefficient and a power of ten. For instance, \(6.50 \times 10^6\) denotes 6,500,000 and is the result of converting 6.50 Tm to Mm.
This method of notation simplifies calculations and makes it easier to compare large and small numbers without excessive zeros, which can lead to mistakes.
Adopting scientific notation across your scientific work ensures clarity and precision, especially in dealing with metric conversions where powers of ten are commonly used.