Question

Perform the following metric-metric conversions. (a) \(5.00 \mathrm{Mm}\) to \(\mathrm{m}\) (b) \(5.00 \mu \mathrm{g}\) to \(\mathrm{g}\) (c) \(5.00 \mathrm{~mL}\) to \(\mathrm{L}\) (d) \(5.00 \mathrm{ds}\) to \(\mathrm{s}\)

Short answer

(a) 5,000,000 m, (b) 0.000005 g, (c) 0.005 L, (d) 0.5 s.

Step-by-step solution

Understand Metric Prefixes

Recognize the prefixes in the given units and their meaning in terms of powers of 10. - Mega (M) means \(10^6\).- Micro (\(\mu\)) means \(10^{-6}\).- Milli (mL) means \(10^{-3}\).- Deci (d) means \(10^{-1}\).

Convert Millimeter to Meter

For part (a), convert \(5.00\) Mm (megameters) to meters. Since \(1\) Mm is equal to \(10^6\) meters:\[5.00 \text{ Mm} = 5.00 \times 10^6 \text{ m} = 5,000,000 \text{ m}\]

Convert Microgram to Gram

For part (b), convert \(5.00\) \(\mu\)g (micrograms) to grams. Since \(1\) \(\mu\)g is equal to \(10^{-6}\) grams:\[5.00 \text{ } \mu\text{g} = 5.00 \times 10^{-6} \text{ g} = 0.000005 \text{ g}\]

Convert Milliliter to Liter

For part (c), convert \(5.00\) mL (milliliters) to liters. Since \(1\) mL is equal to \(10^{-3}\) liters:\[5.00 \text{ mL} = 5.00 \times 10^{-3} \text{ L} = 0.005 \text{ L}\]

Convert Deciseconds to Seconds

For part (d), convert \(5.00\) ds (deciseconds) to seconds. Since \(1\) ds is equal to \(10^{-1}\) seconds:\[5.00 \text{ ds} = 5.00 \times 10^{-1} \text{ s} = 0.5 \text{ s}\]

Key concepts

Metric Prefixes

In the world of measurements, metric prefixes are crucial as they determine the scale of the units we are dealing with. Metric prefixes are a set of standard words that, when added before a basic unit, provide clarity on the size of the measurement. Each prefix corresponds to a power of ten.

Consider the following common metric prefixes:

• Mega (M): This prefix means "million" and corresponds to a factor of \(10^6\). So, 1 Megameter (Mm) is 1,000,000 meters.
• Micro (\(\mu\)): This prefix stands for "millionth," representing \(10^{-6}\). Thus, 1 microgram (\(\mu g\)) equals 0.000001 grams or one millionth of a gram.
• Milli (m): Signifying "thousandth," milli equates to \(10^{-3}\). Hence, 1 milliliter (mL) is 0.001 liters.
• Deci (d): Means "tenth," which gives us \(10^{-1}\). Thus, 1 decisecond (ds) is 0.1 seconds.

Understanding these prefixes is key for properly converting between different units within the metric system.

Unit Conversion

Unit conversion is the process of changing a measure into an equivalent measure that uses a different unit. In metric conversions, it typically involves multiplying by powers of ten. When converting between metric units, be attentive to the prefix and its associated power of ten.

Let's examine how to perform unit conversion using the exercise as an example. If you're asked to convert 5.00 Megameters (Mm) to meters, you multiply 5.00 by \(10^6\), resulting in 5,000,000 meters because 1 Mm equals 1,000,000 meters. This process can be applied to other units as well.

When dealing with smaller units like micrograms to grams, you again multiply by the micro prefix’s power of ten, which is \(10^{-6}\). Therefore, 5.00 \(\mu g\) is equal to \(5.00 \times 10^{-6}\) grams, resulting in 0.000005 grams.

By consistently applying the power of ten associated with each prefix, you can accurately convert one metric unit to another without any confusion.

Powers of Ten

Powers of ten are a fundamental concept in mathematics and are especially important in metric conversions. This concept allows us to express very large or very small numbers in a compact form using exponential notation, which greatly simplifies calculations and conversions.

When we use powers of ten in metric prefixes, it improves precision and clarity. For example, the prefix Mega, which stands for \(10^6\), indicates that any measurement with this prefix will be one million times the base unit. Conversely, the prefix Micro, represented by \(10^{-6}\), means the measurement is one millionth of the base unit.

Using powers of ten also assists in mental calculations, as multiplying or dividing by them shifts the decimal point. Consider converting 5.00 mL to liters: \(5.00 \times 10^{-3}\) moves the decimal 3 places to the left, resulting in 0.005 liters.

Embracing powers of ten in metric conversions not only ensures accuracy but also speeds up the process of converting between units, enabling more efficient problem-solving.