How do I handle units with exponents, like cubic meters?

Apply the conversion factor raised to the same exponent. To convert $m^3$ to $cm^3$, multiply by $(100 ext{ cm} / 1 ext{ m})^3$, which equals $10^6 ext{ cm}^3 / 1 ext{ m}^3$.

Can I chain multiple conversion factors in one calculation?

Yes. Any number of conversion factors can be multiplied in sequence. Because each factor equals exactly one, the actual physical quantity remains unchanged.

How do I deal with standard metric prefixes?

Treat metric prefixes as numerical multipliers. For instance, 'kilo-' means $10^3$. Create a direct conversion factor such as $10^3 ext{ m} / 1 ext{ km}$.

How to convert units with dimensional analysis

Dimensional analysis converts a quantity from one set of units to another by multiplying by a sequence of conversion factors. It applies whenever the relationship between the starting units and the target units is defined by multiplicative constants.

The setup

Identify the given quantity with its initial units and the desired final units. Gather the exact conversion relationships (e.g., 1 inch = 2.54 cm) that link the initial units to the target units.

The steps

Write the initial quantity as a fraction. 2. Create a conversion factor for the first unit to be changed, formatting it as a fraction equal to one. 3. Arrange the conversion factor so the unit you want to cancel appears on the opposite side of the fraction (numerator vs. denominator). 4. Repeat for any remaining units. 5. Multiply the numerical values and cancel the matching units algebraically.

Checking the result

Confirm that all intermediate units cancel perfectly, leaving only the target units. Check that the final numerical value's order of magnitude makes physical sense given the conversion.

Common errors

Inverting the conversion factor (e.g., multiplying when you should divide) is the most frequent mistake. Another common error is failing to apply exponents to both the number and the unit when converting areas or volumes (e.g., converting square meters to square centimeters requires the factor 100^2, not 100).

Worked example

Convert 65.0 miles per hour to meters per second. Given: 1 mile = 1609 m, 1 hour = 3600 s.

Start with the given quantity: $65.0 \frac{ ext{mi}}{ ext{h}}$ . Multiply by the conversion factor for distance: $65.0 \frac{ ext{mi}}{ ext{h}} imes \frac{1609 ext{ m}}{1 ext{ mi}}$ . Multiply by the conversion factor for time: $65.0 \frac{ ext{mi}}{ ext{h}} imes \frac{1609 ext{ m}}{1 ext{ mi}} imes \frac{1 ext{ h}}{3600 ext{ s}}$ . Cancel the 'mi' and 'h' units: $\frac{65.0 imes 1609}{3600} \frac{ ext{m}}{ ext{s}}$ . Compute the final value: $29.0 ext{ m/s}$ .

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References: OpenStax University Physics Volume 1, Chapter 1: Units and Measurement · University Physics with Modern Physics (Young and Freedman), Chapter 1 · Khan Academy: Intro to dimensional analysis