How to Convert Compound SI Units with Squared Denominators

Convert compound units one factor at a time

A compound unit combines units through multiplication, division, or powers. The safest conversion method is factor-label analysis: multiply by ratios equal to one, orient each ratio so an unwanted unit cancels, and leave only the requested units.

For a unit in the denominator, write the old denominator unit in the conversion factor's numerator. That placement lets the matching symbols cancel algebraically.

Exponents apply to the whole conversion factor

When a unit is squared or cubed, both the numerical relationship and the unit must receive that power. Since $1\ \mathrm{m}=100\ \mathrm{cm}$,

$$1\ \mathrm{m^2}=(100\ \mathrm{cm})^2=10{,}000\ \mathrm{cm^2}.$$

Using a factor of 100 instead of 10,000 is the most common error in an area-unit conversion.

A fresh example

To convert $12\ \mathrm{J/(kg\cdot m^2)}$ to $\mathrm{J/(g\cdot cm^2)}$, arrange the factors as

$$12\ \frac{\mathrm{J}}{\mathrm{kg\cdot m^2}} \times\frac{1\ \mathrm{kg}}{1000\ \mathrm{g}} \times\left(\frac{1\ \mathrm{m}}{100\ \mathrm{cm}}\right)^2.$$

The old kilogram and square-meter units cancel, leaving the requested denominator. The numerical multiplier is $10^{-3}\times10^{-4}=10^{-7}$, so the result is $1.2\times10^{-6}\ \mathrm{J/(g\cdot cm^2)}$.

Check the unit algebra before the arithmetic

A reliable check has three parts: every old unit cancels, only the requested units remain, and the power-of-ten change matches the direction and exponent of each conversion. Treating units like algebra catches most compound-unit errors before calculator work begins.